this work addresses the problem of explaining the second law of thermodynamics on the basis of the microscopic laws of mechanics . as discussed earlier the second law of thermodynamicsis commonly understood as composed of two parts which we shall conventionally label as `` part a '' and `` part b '' .`` part a '' is essentially a statement about the _ existence of entropy_. it says that there exists an integrating factor , interpreted as inverse temperature , such that is an exact differential , where is the heat exchanged during a very small variation of the thermodynamic state .this implies that there exists a function of the thermodynamic state , the entropy , that generates the exact differential `` part a '' of the second law evidently pertains to _ equilibrium thermodynamics_. its mechanical foundations have been studied in a previous work .the main conclusion drawn in that work was that the laws of ergodic hamiltonian mechanics alone are sufficient for providing a rather satisfactory explanation of this part of the second law .the approach adopted was that of establishing a correspondence between the _ thermodynamic _ quantities and certain suitably chosen time - averaged _ mechanical _ quantities .thanks to a generalization of helmholtz theorem , then we were able to find the proper _ mechanical analogue _ of thermodynamic entropy .this is the so called _ volume entropy _ : where is the system s hamilton function .this is a function of the dimensional phase - space vector and the external parameter . is an arbitrary constant with the dimensions of action .the present contribution completes the previous one by addressing `` part b '' of the second law .`` part b '' is essentially the law of entropy increase , and as such is a statement that pertains to _ non - equilibrium thermodynamics_. to avoid any possible confusion , here by `` part b '' of the second law and `` law of entropy increase '' , we mean clausius formulation of the entropy principle :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the entropy principle : for every _ nicht umkehrbar _ process in a thermally isolated system which begins and ends in an equilibrium state , the entropy of the final state is greater than or equal to that of the initial state . for every _ umkehrbar _ process in a thermally isolated system the entropy of the final state is equal to that of the initial state _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  the expressions _nicht umkehrbar _ and _ umkehrbar _ could be translated into the current scientific english as _ non quasi static _ and _ quasi static _ respectively .it must be stressed that the entropy principle refers to transformations caused by the variation of some external field , and is not at all a statement about the _ spontaneous tendencies _ of physical systems .if the variation of the external field is acted in such a way as to drive the system out of equilibrium ( non quasi static process ) the entropy will increase .if it is acted in such a way that the system will remain arbitrary close to equilibrium ( quasi static process ) then the entropy will not change .indeed the volume entropy already well accounts for the `` quasi static '' part of the entropy principle . in facts , as it is known since the work of , the volume entropy is an _ ergodic adiabatic invariant_. namely a quantity that does not change during a quasi - static transformation of the external field .as pointed out in , the laws of ergodic hamiltonian mechanics alone provide a quite satisfactory explanation of the `` quasi static '' part of the entropy principle : no statistics is needed to explain the equal sign in the entropy principle .it must be emphasized that here we are establishing a correspondence between the thermodynamic concept of quasi static process and the classical mechanical concept of adiabatic transformation .in particular the expression `` adiabatic '' will not be used as synonym of `` thermally insulated '' as customarily happens in thermodynamics text - books .obviously the fact that the volume entropy well accounts for `` part a '' and the quasi static part of `` part b '' of the second law , suggests that it might turn out to be very useful in addressing the non quasi static part of `` part b '' .simple considerations suggest that the latter could be proved only in some statistical or averaged sense , though .consider for example a 1d particle of mass in a 1d box of length .the particle bounces forth and back inside the box .let be the energy of the particle .imagine that we can change to length of the box by moving the right wall of the box .the volume entropy of this elementary ergodic system is simply .now imagine that we perform a very fast compression of the box , much faster than the particle period of motion .let be the final length of the box .imagine that during this transformation the particle is far from the moving wall and does not bounce against it .its energy would not change but the change of its volume entropy would be . namely it would be negative .this simple argument should convince that any purely mechanical attempt to prove the entropy principle on the basis of volume entropy would be vain .we certainly need to add some statistical ingredient if we want to prove it .thus we are going to assume that the initial energy of our insulated system is not known .all we know is that it is within some range with some probability .the symbol denotes the `` density of states '' at energy . is also named surface integral or structure function . for example , if we first place the system in thermal contact with a heat bath at temperature , and then we remove the contact , we will not know for certain what the energy of the system will be , but we will know that .  in this work we will prove that , provided that is a decreasing function of , the _ expectation value _ of the volume entropywill be larger than its initial one .it turns out that such proof is much easier in quantum mechanics rather than classical mechanics .therefore we shall first quantize the volume entropy and then study its behavior under the action of a varying field , that is a time - dependent perturbation .the paper is organized as follows . in sec .[ sec : qve ] we introduce the quantum counterpart of the classical volume entropy . in sec .[ sec : perturbation ] we prove that the expectation value of such quantum operator can only increase , if the initial conditions is represented by a decreasing ordering of probabilities . in sec .[ sec : kelvin ] we discuss how this result relates to thomson s formulation of the second law , whereas in sec .[ sec : comparison ] we compare the quantum volume entropy with other quantum entropies present in the literature . in sec .[ sec : classical case ] we show how to adapt the quantal proof of sec .[ sec : perturbation ] to the classical case .the role of the initial equilibrium is discussed in sec .[ sec : initial - condition ] .we will discuss the fact that the results proven in the paper are direct consequences of the time - reversal symmetric microscopic laws , and that , besides the law of entropy increase , there exists a law of entropy decrease as well .the conclusions are drawn in sec .[ sec : conclusion ] .let us first consider the 1d case . in 1dthe volume entropy reads : this can be conveniently reexpressed as the logarithm of the _ reduced action _ this is also known as the helmholtz entropy .quantization of the helmholtz entropy is almost immediate . indeed , using a colorful expression , i would say that eq .( [ eq : helment ] ) invites the reader to quantize .using the semiclassical approximation of bohr - sommerfeld and setting equal to plank s constant allows to see that is a quantized quantity whose possible values are ( within the range of validity of the approximation ) : we can extend this reasoning to multidimensional systems whose dynamics is ergodic . in this general casethe volume entropy is given by eq .( [ eq : volumeentropy ] ) . again using the quasi - classical viewpoint the integral in eq .( [ eq : volumeentropy ] ) approximately counts the number of quantum states not above a certain energy .since the levels are non degenerate this number is , where one considers that the vacuum state counts as a half state .the levels are non - degenerate because the corresponding classical dynamics is ergodic .this can be understood by noticing that ergodicity implies that the hamiltonian is the only integral of motion .this , translated into the language of quantum mechanics , says that the hamiltonian alone constitute a complete set of commuting observables , so that the only quantum number is .  at this point , it is quite easy to construct the quantum version of volume entropy .consider a finite ( i.e. , not necessarily macroscopic ) _ non - degenerate _quantum systems .let be the _ quantum number operator _, i.e. : where is the complete orthonormal set of hamiltonian s eigenstates . , the total number of energy levels , can be infinite .the eigenvectors of are the energy eingenvectors , and the eigenvalues are the corresponding quantum numbers .then the _ quantum volume entropy operator _ can be defined as : we adopt a system of units where , boltzmann constant , is equal to .armed with a quanto - mechanical analogue of thermodynamic entropy ( [ eq : s = logn ] ) , we can now study its evolution under a time - dependent perturbation .as prescribed by the entropy principle we shall assume that the system is thermally isolated from the environment .as discussed previously , the system energy is not known .this means that the system is assumed to be in a statistical mixture of states , described by a density matrix , rather than a pure state .as prescribed by the entropy principle we shall also assume that the system be initially at equilibrium .we will translate this thermodynamic notion into the quantal requirement that .so the system is at equilibrium whenever and it is out of equilibrium whenever . at ,we switch on a perturbation .this is implemented by changing the value of some external parameter during the course of time : . can be for example the volume of a vessel containing the system , or the value of some external field like an electric or a magnetic field . at time , the perturbation is switched off .we assume that at some time any transient effect will be vanished and the system attains a new equilibrium state described by some , such that .thus , before time and after , the system is at equilibrium , and for it is out of equilibrium . due to the perturbation the hamiltonian changes from the initial value to the final value , and accordinglythe quantum entropy operator will change in time and move from to .we introduce the following time - dependent orthonormal basis set .the vectors are defined as the eigenvectors of the `` frozen '' hamiltonian .that is : since at time  , then =0 $ ] .this means that is diagonal over the initial basis : as anticipated in the introduction we shall assume that is decreasing : our definition of quantum entropy ( [ eq : s = logn ] ) , is essentially an _ equilibrium _ definition . using the bases , we can extend the definition to the _ out of equilibrium _ case , as following : we shall assume that non - degeneracy is kept at all times .this implies that there is _ no level crossing _ , and ensures that the quantum number operator gives the correct eigenvalues at all times .the same assumption is used by to ensure the proper ordering of energy eigenvalues .note that , unlike the hamiltonian s spectrum ( [ eq : spectum - h ] ) , the spectrum of the quantum entropy ( [ eq : spectrum - s(t ) ] ) is time - independent .we define the transition probabilities : where is the time evolution operator expressed in terms of the time - ordered exponential .the s represent the probabilities that the system will be found in the state at time provided that it was in the state at time .they satisfy the relations : and for the change in the expectation value of the quantum entropy of eq .( [ eq : s = logn ] ) we have : - tr \left[\rho_i   \mathcal{s}_i\right ] =  \sum_{n=0}^k ( p'_n - p_n ) \ln \left(n+\frac{1}{2}\right)\ ] ] where is the probability that the system is in state provided that the initial probabilities were .  using the `` summation by parts '' rule : eq .( [ eq : sf - si ] ) becomes we have : from eq.s ( [ eq : sum - a - nk ] ) and ( [ eq : a - nk>0 ] ) we have and , therefore using the ordering of probabilities ( [ eq : orderingprobabilities ] ) we get ( see also ) : where we used eq .( [ eq : sum - a - nk ] ) in the last line .noting that in eq .( [ eq : deltas ] ) , we finally reach the conclusion that : this inequality holds for any transformation acted on a thermally insulated , non degenerate quantum system which is initially at equilibrium with a decreasing ordering of probabilities . to complete the proof of the entropy principle we have to prove that the equal sign holds for adiabatic transformation .note that the non - degeneracy assumption ensures that the _ quantum adiabatic theorem_ holds .this ensures that the transition probability between states with different quantum number will be null during an adiabatic transformation : therefore , for an adiabatic transformation we get ( see eq .[ eq : pn ] ) , which brings to this concludes our quanto - mechanical proof of the entropy principle .note that the result in eq .( [ eq : s - f = s - i ] ) is not surprising because the quantum entropy operator has been defined as the quantum counterpart of a classical adiabatic invariant .  also note that we have established and used the following correspondences between thermodynamics and quantum mechanics :  * entropy * equilibrium * ( non)quasi - static process (non)adiabatic perturbation  since the equilibrium condition for the final state has never been used in the proof , inequality ( [ eq : entropy - increase ] ) holds for any .note that this by no means implies that \ ] ] is a monotonic increasing function of time .all we can say is that if at times the density matrix is diagonal and its spectrum is monotonic decreasing , then : in general there can well be two times , where for example the system is out of equilibrium , such that .it is important to stress that , when the system is out of equilibrium the quantity should nt be regarded as the system s thermodynamic entropy , which is essentially an equilibrium property .thus is only one of the many possible out of equilibrium generalizations of entropy .what makes it special is that it proves effective in addressing the entropy principle .the present proof of the entropy principle is very close , in the approach and methods , to a result discussed recently by .they considered the following alternative formulation of the second law , which they attribute to kelvin ( w. thomson ) :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thomson s formulation : no work can be extracted from a closed equilibrium system during a cyclic variation of a parameter by an external source ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  if we denote the work done by the external source as , the principle can be expressed simply as : the proof of goes like the one we have proposed above for the entropy principle .indeed that work has been a major source of inspiration for the present one . in this case one wants to study the following quantity : - tr [ h_i\rho_i]\ ] ] for a cyclic process .this means that the final hamiltonian is assumed to be equal to the initial one .thus : where are the eigenvalues of .these are ordered according to .the eigenvalues , play here the same role as the entropy eigenvalues , in eq .( [ eq : sf - si ] ) .thus it is immediate to see that , under the assumption of decreasing probabilities ( [ eq : orderingprobabilities ] ) , eq .( [ eq : w=0 ] ) holds quanto - mechanically .  in a subsequent work have extended this result to the case of possibly non cyclic transformation .they have found that where are the eigenvalues of the final hamiltonian , is the work actually performed on the system and is the work that would have been performed if the same transformation would have been carried adiabatically .the proof is formally equivalent to the one discussed here .( [ eq : w > tildaw ] ) expresses the minimal work principle according to which , whenever we perform a non - adiabatic transformation , we spend more work than we would have if performing an adiabatic one .  the formal similarity of eq .( [ eq : w > tildaw ] ) and eq . ( [ eq : sf - si ] ) proves that the formulation of the second law as a principle of minimal work or as an entropy principle are _ equivalent_. in particular it is easily seen that : thus the two principles are equivalent .further , whenever one is violated , the other will be too .cases where the minimal work principle is violated because of level crossing are discussed by . in those casethe entropy principle would be violated too .  the history behind this kind of quanto - mechanical proofs of the second law is relatively recent , and can traced back at lest to the works of and . due to the lack of a suitable quanto - mechanical analogue of entropy , though , the application of such quantal approaches has remained restricted to the analysis of statements that concern work , whose mechanical definition is quite straightforward . to the best of the author s knowledge , similar arguments and approaches have been previously proposed for addressing the entropy principle only in the relatively un - known work of . already proposed the quantum entropy operator in the form , but the connection with the generalized helmholtz theorem ( which has been introduced later ) was not made , neither the importance of the volume entropy as a good mechanical analogue of thermodynamic entropy for possibly low dimensional systems was recognized .unlike the present work , in fact , the work of is concerned only with the macroscopic case . ) and eq .( [ eq : w > tildaw ] ) .it is interesting to notice that did not published his result because eq . ( [ eq : w > tildaw ] ) was proven previously by . to the best of my knowledge , the proof of eq .( [ eq : entropy - increase ] ) based on the logarithm of the principal quantum number was never given before though , thus it remained unpublished . ]the employment of the quantum volume entropy improves quite a lot over previous attempts at explaining the entropy principle based on quantum entropies .in fact , the employment of the entropy in eq . ( [ eq : non - eq - entropy ] ) has many advantages over other quantum mechanical entropies present in literature .in contrast with von neumann entropy : .\ ] ] the expectation value of the quantum operator does change in time , and it has been proved to increase under the assumption discussed .tolman s coarse - grained entropy : does change in time and it is an adiabatic invariant .nonetheless it fails in accounting for the inequality sign in the case of non - adiabatic perturbations .all we known is that for an infinitesimal _ abrupt _ transformation that begins and ends in a _canonical equilibrium _the corresponding change in is non - negative .but this does not ensure that for any _ finite _ non - adiabatic transformation the change would be non - negative as required by clausius formulation .tolman s argument according to which any finite transformation could be reproduced by a sequence of many infinitesimal abrupt transformations each followed by the reaching of a canonical equilibrium does not seem to be tenable .in fact , as a result of a finite non - adiabatic transformation , the system could well end up in a non - canonical distribution .further , tolman s definition of entropy of eq .( [ eq : cgent ] ) applies only to macroscopic equilibrium systems . on the contrary the result proved in sec .[ sec : perturbation ] holds _ no matter the number of degrees of freedom _ of the system .thus the quantum volume entropy might turn out to be very useful in the novel and fast growing field of _ quantum thermodynamics _ of nanoscale systems .see for example , see also .the result of eq . ( [ eq : entropy - increase ] ) can be proved also classically .let the system be initially distributed according to some probability distribution function .let be the final distribution .let and denote the volumes enclosed by the hyper - surfaces and respectively . where and are the initial and final hamiltonians .the volume entropy of a representative point that at time lyes on the hyper - surfaces is .we have a similar expression for time .then : where denotes the initial ( ) or final ( ) , structure function .note that in general thus we can make the change of variable in the integrals .let , then we have ( after dropping the subscript in ) : this is the classical analogue of eq .( [ eq : sf - si ] ) .the role of is played by the `` enclosed volume '' , and the discrete probabilities are now probability density functions . since the evolution is deterministic , it is possible to express the final probability in terms of the initial one as where is the green function associated to the evolution of probabilities in space .that is represents the evolved at time of a dirac delta centered around at time .if we denote the time evolution operator that evolves probabilities in space from time to time as , is defined as : the function is the classical counterpart of the transition probability .evidently , thanks to the classical adiabatic theorem we have for an adiabatic switching : this is the classical counterpart of eq .( [ eq : a - kn = delta - kn ] ) . for non adiabaticswitching we expect , considered as a function of , to be bell - shaped with some finite width .the problem of determining the shape of has been studied by , who proved that , within the second order of adiabatic perturbation theory , actually drifts and diffuses according to an effective fokker - planck equation .since represents a probability distribution function in space , it satisfies : and using liouville s theorem it is also possible to prove that : the analogy with the quantum case has been completely established now , and the proof of the entropy principle follows by repeating the same steps .the requirement on the initial distribution is that it be a decreasing function of .since is increasing , this requirement translates into the requirement that be a decreasing function of .inequality ( [ eq : entropy - increase ] ) holds as a direct consequence of the time reversal symmetric microscopic laws of quantum or classical mechanics . as such, it does not entail any arrow of time .the reason for the emergence of the sign in eq .( [ eq : entropy - increase ] ) should be looked for , rather , in the fact that we have considered only a certain restricted subset of all possible initial conditions .to explain this point it might be useful to see our ensemble of systems as a box containing many balls ( see fig . [fig : shake ] ) .( i.e. the entropy ) would increase.,width=491 ]  each ball represents an element of the ensemble .the box is divided into labelled cells that represent the quantum states .the cell most close to the left wall is state with , its right neighbor cell is the state and so forth . at time , the balls are distributed in the box according to some probability .we can see the time dependent perturbation acted on the system as the action of shaking the box .the effect of the shaking is that of flattening out the initial distribution .thus if initially we had some accumulation of balls towards the left side of the box , we expect the final state to be more flat . if we look at the average value of or any other increasing function of , like for example , we would record an increase of such values .this is a mere consequence of the fact that initially we had an accumulation towards the left .if initially we have had an accumulation towards the right , again the shaking would flatten out the distribution , but this time we would see a decrease of the average value of and of . if instead the initial distribution were flat , we would see no change in those quantities .indeed it is quite easy to see that the sign of inequality ( [ eq : entropy - increase ] ) would be reversed if an increasing ordering of initial probability is assumed .therefore , for such subset of the set of all possible initial distributions , we actually have a law of entropy decrease !this reflects the fact that there is no asymmetry in the time evolution of the volume entropy operator .thus , in principle , it should be possible to observe a decrease of entropy if the initial equilibrium would be given by an _inverted population_. in other words , we should be able to observe an inverted second law of thermodynamics in _ negative temperature _ systems .indeed experimental evidence of this exists since the very pioneering works of pound , purcell and ramsey on spin systems .they observed that  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` when a negative temperature spin system was subjected to resonance radiation , more radiant energy was given off by the spin system than was absorbed '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  this means that it is possible to extract work from a negative temperature system by means of a cyclic transformation .in other words , for negative temperature systems we already have experimental evidence that : because of the _ equivalence _ of the minimal work principle and the entropy principle , in this case we would also have : the fact that the law of entropy increase is overwhelmingly more often observed than its symmetrical law of entropy decrease is a consequence of the fact that positive temperatures are overwhelmingly more common than negative ones .the former in fact is the _ natural _ state of matter , whereas the second can only be created _ artificially _ and only in few very special cases . already pointed out that very strict conditions must be met for a system to be capable of negative temperatures : ( a ) the system must be at equilibrium ( b ) there must be an upper limit in the hamiltonian s spectrum ( c ) the system must be thermally isolated from the environment .the second requirement is very restrictive as most systems have an unbounded kinetic energy term in the hamiltonian .also the requirement ( c ) is restrictive in the sense that thermal insulation can be achieved only approximately and for a certain amount of time . on the contrary the inevitable thermal contact of our system with its environmentwould eventually bring it to the monotonic decreasing gibbs state the latter describes the natural state of matter , and as such is the _inevitable _ initial condition of any thermodynamic experiment .thus the time asymmetry of the laws of thermodynamics arises at the level of the initial thermal equilibrium , rather than in the second law itself .this seems to be in agreement with the view expressed by , according to which the second law does not entail any time asymmetry .the origin of the arrow of time should be looked for , instead , in the minus first law of thermodynamics , namely the _ equilibrium principle _ .adopting an approach similar to those adopted previously to prove the thomson s formulation of the second law , here we have proved the entropy principle on the basis of quantum mechanics , for initial conditions characterized by decreasing probabilities .this completes a programme devoted to the study of the mechanical foundations of thermodynamics initiated with a previous work . that work addressed the equilibrium part of the second law ,whereas the present one addresses the out of equilibrium part .the key tool of investigation adopted in both studies is the volume entropy . herewe have compared it to other quantum entropies and shown that it proves more effective in addressing the entropy principle .we discussed the equivalence of the entropy principle and thomson s principle and have seen that the entropy principle can be proved classically as well . the apparent time asymmetry expressed bythe second law stems from the fact that initial decreasing probabilities are overwhelmingly more common and natural in ordinary experimental set - ups than increasing ones . indeed a reversed entropy principle can be observed in artificially prepared negative temperature systems .

Abstract: in a previous work ( m. campisi . stud . hist . phil . m. p. 36 ( 2005 ) 275 - 290 ) we have addressed the mechanical foundations of equilibrium thermodynamics on the basis of the generalized helmholtz theorem . it was found that the volume entropy provides a good mechanical analogue of thermodynamic entropy because it satisfies the heat theorem and it is an adiabatic invariant . this property explains the `` equal '' sign in clausius principle ( ) in a purely mechanical way and suggests that the volume entropy might explain the `` larger than '' sign ( i.e. the law of entropy increase ) if non adiabatic transformations were considered . based on the principles of microscopic ( quantum or classical ) mechanics here we prove that , provided the initial equilibrium satisfy the natural condition of decreasing ordering of probabilities , the expectation value of the volume entropy can not decrease for arbitrary transformations performed by some external sources of work on a insulated system . this can be regarded as a rigorous quantum mechanical proof of the second law . we discuss how this result relates to the minimal work principle and improves over previous attempts . the natural evolution of entropy is towards larger values because the natural state of matter is at positive temperature . actually the law of entropy decrease holds in artificially prepared negative temperature systems .  quantum adiabatic theorem , minus first law , negative temperature , minimal work , helmholtz theorem , arrow of time .


in recent years complex networks have attracted much of interest to model real - life networks such as social , biological and communication networks .certainly , introducing the essential static and dynamic features of these networks can help us in a better understanding of their various properties .a well known property of most of these networks , the so called small world property indicates that the average distance between any two nodes increases slowly with the size of the network ( i.e. as logarithm of the size ) .this in turn can lead to a fast spreading of effects in the network and so increases the finite size effects when one studies for example diffusion in such a network .extensive studies also indicate the importance of degree ( the number of neighbors of a node ) distribution for static and dynamic behaviors of the network .one can also add various kinds of correlations , e.g. degree correlation of two neighbors , to the list of these important features .+ protein interaction networks are important examples of the real - life networks in which nodes and edges represent proteins and interactions between them respectively .proteins have been traditionally recognized on the basis of their roles as enzymes , signalling molecules or structural components in cells and micro - organisms .the most rudimentary structural information about the proteome ( assembly of proteins in an organism ) is the pattern of interactions between different proteins .determining such connections , helps us in understanding the backbone of functional relationships between proteins and the pathways for the propagation of various signals among them . besides specific detailed information about the pattern of interactions in a single proteome , which are certainly important for its functioning , some general characteristics of these networks are also important in that they may point to universal properties of organisms .for example , it has been shown that as far as the interaction of individual proteins are concerned , the interaction network of the budding yeast _ saccharomyses cerevisiae _ ( _ s .cerevisiae _ ) is a scale free network .this property is itself a hint to the robustness of the protein interaction network against the random removal of proteins .+ however recent progress indicates that each of the central processes in a cell _ is catalyzed not by a single protein but by the coordinated action of a highly linked set of several proteins , called a protein complex _ .thus complexes act as protein machines and have been evolved for the same reason that humans have invented mechanical and electronic machines .it is also remarkable that a single protein may be shared in several different complexes .it is expected that a protein with a general functionality is shared by many complexes . on the other hand it seems that the weight of a complex depends on the degree of sophistication of its tasks .+ it is in view of this new emerging picture of the proteome as a coordinated ensemble of protein complexes that we study the general properties of the network of protein complexes of the budding yeast .certainly , analysis of the proteome map at both protein and complex level will result in a better understanding of the proteom functioning .although from a biological point of view the precise nature of proteins and their interactions in a single living organism are important , from a physical point of view in which we seek the general universal patterns among many different organism , we can study the most elementary features of a proteome , i.e. the weight distribution of complexes , distribution of the number of proteins shared between two complexes , etc .+ based on extensive information provided in we constructed a weighted graph corresponding to the network of complexes . by thiswe mean that both the nodes and the edges are assigned weights .the weight of a node is equal to the number of proteins in the complex it represents .two nodes are connected by a weighted edge , whose weight indicates how many proteins are common in the two complexes .such a point of view seems to be the first step to understand the integration and coordination of cellular functions .connections in this network not only reflect physical interaction of complexes , but may also represent common regulation , localization , turnover or architecture .in addition it has a meaningful interpretation in other real - life networks such as social networks when a connection between two communities can only be established by common individuals in them .note that one may consider a single protein complex as a subgraph of the protein interaction network with a high level of interconnection between its elements .from this point of view the protein complex network is a large scale or coarse grained picture of the protein interaction network .however , we stress that in view of the recent findings , this is not the correct picture for the interactome .in fact the important feature of the recent experiments compared to the previous experiments is that they uncover not only the pairwise interactions but ternary , quaternary and higher interactions between different proteins in a complex .+ in this work we show that the above protein complex network is a small world one with scale free degree distribution. moreover it is found that distributions of weights of complexes , weights of edges and coordination numbers of proteins ( the number of complexes a protein participates ) follow power law behaviors which can in turn refer to a kind of preferential attachment in the evolution of the protein complex network .we compare the network with some null models such as erds and rnyi random graph and a random - selection model in which each complex selects its proteins randomly from the list of all proteins . unlike the former case the latter model results in a considerable clustering for the network of complexes . here by clustering we mean the probability that two neighbors of a node be also connected to each other . in this mannerthe clustering is equivalent to the transitivity of the network ( defined as three times the ratio of the number of triangles in the network to the number of connected triples of nodes ) .moreover , we find that the random - selection model can well reproduce the degree distribution of complexes and the dependence of degree of a complex on its weight .however some important distributions of the network such as weight of edges are still far from the predictions of this model in the region of large weights . following the previous models for the evolution of the protein interaction network , we propose a simple evolutionary model based on duplication and divergence ( mutation ) of proteins .we show that this model reproduces the power law behavior of some essential distributions such as weight of complexes and weight of edges .the paper is organized as follows .section ( [ 2 ] ) is devoted to the description of the budding yeast protein complex network based on the data provided in .section ( [ 3 ] ) gives a comparison between the random - selection model and the real network .the evolutionary model is introduced in section ( [ 4 ] ) .we conclude the paper in section ( [ 5 ] ) .according to the data we have extracted from , the _ s. cerevisiae _ includes proteins , organized in complexes .the resulting network is then composed of nodes and edges .a complex of weight is denoted by a node of weight , and if two complexes share proteins , a weight is assigned to the edge connecting them .figures ( [ f1]-a ) and ( [ f1]-b ) show the distribution of weights of the nodes , , and weights of the edges , .    both distributions show the power law behavior ( and ) with exponents respectively equal to and , figures ( [ f1]-a and [ f1]-b ) .the observed deviation in the number of complexes of weight from the expected power law behavior could be mostly attributed to the experimental limitations .the average weight of the nodes and the edges turn out to be respectively and with the following dispersions : and .one could attribute the large values of these dispersions to the small size of the network and scale free nature of the related distributions .a curious property of the network is that there is no correlation between the weights of adjacent nodes . in other words , the probability that an emanating edge from a complex of weight encountersanother complex of weight is independent of . to show this we compute the associated correlation coefficient , , which in the protein complex network has the value andis defined as follows : let be the probability that an arbitrary edge lies between two complexes of weights and . thus gives the probability of finding a complex of weight at the end point of an arbitrary edge .now the correlation coefficient is given by it is clear that in the absence of any correlation ( i.e. when ) we have .if similar complexes have a high tendency for being connected to each other we say that the network is assortative in this respect and the correlation coefficient will be positive .otherwise the network is dissortative and the correlation coefficient will be negative .obviously one can apply this definition to any two point distribution to measure the degree of correlation between the two variables .+ one may also ask what is the relation between the degree of a complex and its weight . our results again give a power law dependence for large values of complex weights .the total weight of edges emanating from a complex of weight ( its weighted degree ) scales as , where , figure ( [ f2]-a ) .it is found that the number of neighbors behaves in a similar way with , figure ( [ f2]-b ). obviously grows faster than with the weight of complex as expected . roughly speaking , since , the above relations suggest that the average weight of an edge emanating from a complex of weight scales as .+ to address topological properties of the graph we studied also the degree distribution , and the correlation between degrees of neighboring nodes which can again be detected by computing the related correlation coefficient , .note that by degree of a node we mean the number of neighboring nodes , irrespective of the weights of the edges emanating from that node . by weighted degree of a node we mean the sum of weights of the edges emanating from this node . in figure ( [ f3 ] )we show that the degree distribution is scale free with a sharp cutoff at its tail . in this figurewe see also a power law fit of form with exponent to the real data . moreover , we find the value for which means that in contrast to the protein interaction networks , there is no strong correlation between the degrees of adjacent nodes in protein complex network .our results shows that the same conclusion is true if we take also the weights of the edges into account .+    usually a given protein takes part in more than one , say complexes .we call the coordination number of that protein .we found that on average a protein takes part in complexes ( with ) .figure ( [ f4 ] ) shows the number of proteins versus their coordination numbers , . the distribution is a scale free one for large coordination numbers , that is with as its exponent .this indicates that there are a few number of proteins with high coordination numbers .it seems that these are the proteins with exceedingly important role in the functioning of the cell .let us compute the correlation coefficient which measures the correlation between coordination of proteins and weight of complexes they contribute . to this endwe consider the collection of proteins and complexes as a bipartite network in which nodes of the first type represent complexes and those of the second type represent proteins .an edge in this network connects a node of the first kind to a node of the second kind .thus the number of edges emanating from a protein determines its coordination number and similarly the number of edges connected to a complex gives its weight .now we define as the probability that an arbitrary edge of this bipartite network connects a complex of weight to a protein of coordination number .therefor and are respectively the probability that an arbitrary edge reaches to a complex of weight and a protein of coordination number .now the associated correlation coefficient is defined as   we find that in the the case of our data which indicates to the absence of correlation in this respect . that is the coordination number of a protein does not affects in its relation with complexes of different weights more than what is expected by chance .+ the network of complexes also defines pathways for the propagation of various signals such as phosphorylation and allosteric regulation of proteins . for such functions ,a key parameter of the network is its diameter , defined as the shortest path between the remotest nodes in the giant component of the network .our analysis reveals that the network has a small diameter , , which points to the small world property of the network . to confirm this we also computed ( where denotes the shortest distance between nodes and ) and compared it with the corresponding quantity in an equivalent erds - rnyi random graph , table ( [ t ] )the reason behind the computation of is that the above graph is not connected rather it has a giant component of size , a binary component and single nodes . to have a measure of clustering in the network ,its transitivity was extracted and it has been found that it is almost six times greater than the one in an erds - rnyi random graph , see table ( [ t ] ) .this high level of transitivity which is another hint to the small world property of the protein complex network would give rise to the robustness of the network against the random removal of nodes or edges .to highlight the special features of the yeast protein complex network and to possibly draw conclusions of biological interest , one may compare it to a random network in which proteins aggregate randomly to form different complexes .we call such a model a random - selection model . the simplest such model may be constructed as a bipartite network as follows : one takes a bipartite network consisting of two types of nodes .nodes of the first type represent complexes and those of the second type represent proteins .a bipartite network of this kind consisting of complexes and proteins and the resulting weighted network consisting of only protein complexes are depicted in figure ( [ f5 ] ) .complexes and proteins .connections can only be established between proteins and complexes .( b ) the resulted weighted graph of protein complexes .the number of lines connecting two complexes represents the number of shared proteins.,width=302 ]  here we start from a bipartite network and calculate many of the properties of the resulting protein complex network exactly .a given complex contains a number of proteins which we denote by . in general . from the real data one can infer the actual sequence .one thus assigns free ( unconnected ) stubs to the first type of nodes ( the complexes ) according to this sequence and then connects these stubs randomly to the second types of nodes ( proteins ) .each protein connected to a complex means that it is contained in that complex . in this way oneobtains another distribution , where is the number of complexes a given protein participates in .clearly a given protein may be contained in more than one complex .note that a peculiar feature of this model is that certain proteins may not be connected to any complex at all .later in this section we introduce another slightly improved model in which this effect does not happen .+ in the following we calculate many of the interesting quantities of the resulting weighted network of complexes exactly and compare them with real data to check the viability of this model .to proceed let denotes the number of complexes of weight .obviously , where is the total number of complexes .first let us calculate the probability of two complexes of weight and to have proteins in common .denote this probability by .the first complex chooses its members freely from the collection of all proteins .the second complex , has ways for choosing its members from the collection , of which ways are available for choosing members in common from the first complex and ways are available for choosing the remaining set disjoint from the first complex .hence the probability is .this equation can be rewritten in the form to make its symmetry under interchange of and manifest .the probability that two such complexes have no common members ( be not connected to each other in the network ) is : for one can approximates by and by .so the above relation takes the form thus the probability of two complexes being connected will be and the average number of edges will be : moreover the average number of edges with weight is given by : figure ( [ f6]-a ) shows this quantity for a model network with the same parameters as the yeast protein complex network .we see that the weight distribution of edges decreases exponentially in contrast to the power law behavior of the protein complex network .this means that in the yeast protein complex there are edges with very high weights .these high weight edges inevitably connect high weight complexes .thus in the yeast complex network , the hubs ( complexes with high degree ) are connected to each other intensively .this is in contrast with the protein interaction network where the high degree proteins have a larger tendency to be connected to low degree ones .+ in the same way one obtains , the average number of neighbors of a given complex of weight : in figure ( [ f6]-b ) we compare the above expression with the one obtained from the real network .one finds a good agreement between predictions of the model and the real data . from the distribution of weights of complexes and the above relation , one can obtain the distribution of degrees using the relation .thus we expect that degree distribution of the random - selection model to be also close to the real one . + what can be said about , the probability that a given protein contributes in complexes ?it is not difficult to show that due to the fully random nature of the selection process , the above distribution has a binomial form like where is the average weight of complexes .thus for very large values of and we have therefore we get a poisson distribution for coordination numbers of proteins in distinctive contrast to the real distribution which is scale free . +as seen from figure ( [ f6 ] ) , our analytical treatment of this model revealed that some of the general characteristics ( the degree of a given complex as a function of its weight and so the degree distribution of complexes ) of these networks are close to those of _ s. cerevisiae _ protein complex network .further results derived from numerical simulations given in table ( [ t ] ) show that random - selection model is rather close to the yeast protein complex network .+ note however that certain discrepancies between the random - selection model and the real complex network , i.e. in the distribution of the weights of edges , persist even if we improve this model by fixing the coordination number of proteins from the beginning to be the same as the one derived from real data of the protein complex network ( see figure ( [ f7 ] ) ) .the evolutionary model that we introduce in the next section is aimed to remove this discrepancy .in this section we introduce a simple model aimed at representing the evolution of protein complex network .the model is based on the extension of the hypothesis of evolution by duplications and divergence of proteins to the level of protein complexes .dna duplication has long been known as an important factor in the evolution of genome size .this process almost certainly explains the presence of large families of genes with related functions in biological complex organisms .consider a set of proteins which belong to a number of functional units or complexes . in each evolutionary step , the following actions take place .a protein is randomly chosen and duplicated .it means that a new protein is added to the proteome and contributes to all the complexes in which its mother participates. then the new protein undergoes mutations ; it loses its membership in any given complex with probability , and enters any other complex with probability .it is also probable that the protein creates a new complex ( novel functionality ) by its own with probability . in figure ( [ f8 ] )we have shown what happens in a typical step of network evolution .has been duplicated and edges of the new protein ( protein ) undergone mutations.,width=302 ]  note that during this process the new protein may not contribute to any complex and thus leaves the proteome .starting from one protein in one complex , we have taken evolutionary steps .we found that it was the best number of steps which could intimately produce the desired results .we found that in this situation only about percent of duplicated proteins are stuff and the others takes part at least in one complex .having we will also have the probability of creating a new complex in each step , because the number of complexes satisfies .the other parameters can also be determined by requesting that the number of proteins and some important distributions ( e.g. weight distribution of complexes ) be as close as possible to the real data .moreover , note that we also expect that the probability of entering a new complex for a duplicated protein must be much smaller than the probability of exiting one of its inherited complexes . summing up these points we find , by fitting to the real data , the following values for parameters of the model : , and .one can find results of this model in figure ( [ f9 ] ) and table ( [ t ] ) . +    in figure ( [ f9]-a ) we see that the model closely reproduces the scale free behavior of the weight distribution of complexes in the region of large weights although there are some deviations from the real data for the number of small complexes .indeed we expected such a power law behavior in advance due to the presence of preferential attachment in the process of evolution of complexes .note that although the selection of proteins for duplication is a completely random procedure , complexes with higher weights have a higher chance to include the new duplicated protein . in this way the evolution of complexes is governed by the well known mechanism of preferential attachment which is a principal way to produce scale free behaviors in the realm of complex networks .+ figure ( [ f9]-b ) shows the weight distribution of edges in the resulted protein complex network of the evolutionary model .the agreement with the real data is excellent .it is one of the essential features of protein complex network which none of the random models studied in this paper could exhibit it .we believe that this fact indicates to the essential role of duplication and divergence of proteins in the evolution of the protein complex network .+ moreover as table ( [ t ] ) shows the evolutionary model results in a small world protein complex network .both the transitivity and diameter of the model are comparable with the values of the real network of complexes .+ this model also exhibits a little tendency for being a negative correlated network in weights and degrees of neighboring complexes .indeed the related correlation coefficients turn out to be and .it means that complexes with different weights and degrees have a larger probability to be connected to each other .+ finally it is worthwhile to note that the data studied in this paper are those of the yeast proteome whereas the results given in this section are the average behavior of such a proteome .to summarize , we have shown that the protein complex network of the yeast is a nearly uncorrelated small world scale free network .the power law behavior was also found in other significant distributions such as weight of complexes , weight of edges and coordination number of proteins .although some of these features ( e.g. small world property and high clustering ) were expected in advance , this study revealed some distinctive properties of this network like the scale free behavior of weight distribution of edges .we also compared the yeast protein complex network with a random - selection model and a simple evolutionary model .it was found that the random - selection model can satisfactorily reproduce the relation between weight and degree of a complex and also degree distribution of the real network .however this model failed to give the power law behavior of the distribution of weights of edges , a property which could be well reproduced in the evolutionary model . in the latter modelthe desired distributions automatically arise by just fitting the model parameters using the real data . + from the evolutionary point of view , the above study can also be a hint to the essential role of duplication and divergence processes in the evolution of proteome and this study is indeed an extension of this mechanism to the level of protein complexes .certainly results of the previous studies on protein interaction network along with the investigation of large scale properties in the protein complex network will help in a better understanding of the general behaviors of such systems .+                               .comparison of the yeast protein complex network ( pcn ) with some models : erds and rnyi ( er ) random graph , the random - selection model ( rs ) , improved random - selection model and the evolutionary model .the first column represents : number of complexes ( n ) , number of proteins ( n ) , average of degree ( ) , weighted degree ( ) , coordination number of proteins ( ) , transitivity ( t ) , diameter of network ( d ) and inverse of shortest distance between two arbitrary nodes ( ) .values in parenthesis refer to dispersion of associated quantity .statistical errors have been denoted where we have done averaging over different realizations .[ cols="^,^,^,^,^,^",options="header " , ]

Abstract: the budding yeast _ saccharomyces cerevisiae _ is the first eukaryote whose genome has been completely sequenced . it is also the first eukaryotic cell whose proteome ( the set of all proteins ) and interactome ( the network of all mutual interactions between proteins ) has been analyzed . in this paper we study the structure of the yeast protein complex network in which weighted edges between complexes represent the number of shared proteins . it is found that the network of protein complexes is a small world network with scale free behavior for many of its distributions . however we find that there are no strong correlations between the weights and degrees of neighboring complexes . to reveal non - random features of the network we also compare it with a null model in which the complexes randomly select their proteins . finally we propose a simple evolutionary model based on duplication and divergence of proteins .


the sensitivities of second generation gravitational wave detectors such as advanced ligo and advanced virgo are expected to be limited by the thermal noise of the test masses within a significant range of signal frequencies around 100 hz  . to reach even better sensitivities ,it has been proposed to use laser beams with an intensity pattern other than that of the fundamental gaussian beam to reduce the effects of this thermal noise  .a beam whose intensity is distributed more homogeneously over the mirror surface , for the same clipping losses , benefits from a more effective averaging over the mirror surface distortions caused by thermal effects  .the specific advantage of using higher order lg modes , as opposed to mesa  and conical  beams , is that they are compatible with spherical mirrors as currently used in gw detectors and other high precision optical setups .research into the potential of the lg mode in gravitational wave detectors has been carried out using numerical simulations and table - top experiments  .the sensing and control signals for an lg beam were found to perform as well as for the fundamental mode in all aspects examined and the lg behaved as expected in short linear and triangular optical cavities .however , an optical cavity resonant for a higher order gaussian mode is degenerate so that a number of modes can resonate at the same time .this is a fundamental difference to a well designed cavity for the fundamental gaussian mode , in which any resonant enhancement of other modes can be suppressed .this degeneracy can potentially cause additional optical losses .simulations have shown that the use of the lg beam , compared to the fundamental mode , lg , could result in a significant contrast defect at the dark fringe  .it is the aim of this paper to investigate how mirror surface distortions affect the purity of an lg beam in high finesse cavities , by analytical calculation and numerical simulation .we will focus on the direct coupling from a distorted mirror and how this affects the mode content in a linear cavity , with our final aim to produce specifications for the mirror surfaces .ideally the mirrors in gravitational wave detectors should be perfectly smooth with a radius of curvature matching that of the incident beam . however , real mirrors deviate from a perfect surface , altering the beams which interact with them. if a beam is incident on a distorted surface described by and uniform reflectivity , then the reflected beam is given by : fig . [fig : coupling_diagram ] illustrates this effect .  .the surface is defined as a height field across a plane perpendicular to the optical axis along . ]  in order to investigate the effects of surface distortions measured mirror surface maps can be used .the term _ mirror map _ refers to an array of data detailing the optical properties of a mirror , often its surface height in nanometers .this data can be used to represent realistic mirrors in numerical simulations of gravitational wave detectors .mirror maps have been produced from uncoated advanced ligo mirror substrates which represent the best mirror surfaces of this kind currently available . in the followingwe have made use of one such map , the surface map of the substrate etm08  . the deviation of this map surface from a perfectly spherical surface with radius of curvature 2249.28 m is shown in fig .[ fig : etm08 ] .this substrate shows an rms surface figure error of 0.523 nm .zernike polynomials are well suited for the purposes of describing mirror surface distortions .zernike polynomials can be used to describe classical distortions such as tilts and curvatures  .they are a complete set of functions which are orthogonal over the unit disc and defined by radial index , , and azimuthal index , , with . for any index have one odd and one even polynomial  : where is the normalised radial coordinate , is the azimuthal angle , is the amplitude and is the radial function .the radial function is given by the following sum : for even and 0 otherwise .this gives non - zero zernike polynomials for each value of ( for the odd polynomial is zero ) .[ fig : zern ] shows the surfaces described by the zernike polynomials corresponding to orders ( n ) 0 to 4 .the lower order polynomials represent some common optical distortions , some of which are summarised in table [ table : zernike ] ..summary of some common names for the lower order zernike polynomials  . [ cols="^,^,^",options="header " , ]   these amplitude limits were applied to the etm08 mirror map , resulting in coupling of 19ppm into modes other than lg and 0.043ppm into the other order 9 modes .the cavity defined in sec . [ sec : ligo_cavs ] was simulated with this limited map , resulting in 815ppm impurity in the circulating beam .this is a very good improvement on the original impurity of 0.114 , illustrating that a high beam purity is achievable with these mirror requirements . to achieve an even higher beam puritywill involve reducing the amplitudes of these polynomials further , as well as additional zernike requirements .we have investigated the coupling which occurs when laguerre - gauss modes are incident on a mirror with surface distortions .taking an analytical approach we used zernike polynomials to represent mirror surface distortions and derived an approximate equation for the significant coupling when an lg mode is reflected from a surface defined by a particular zernike polynomial .this derivation resulted in a condition for significant coupling , , where is the azimuthal index of the zernike polynomial and and are the azimuthal indices of the incident and coupled modes respectively .this is a significant result as it allows us to predict which order 9 modes will be largely coupled by particular zernike polynomials and suggest which modes will have large amplitudes in the arm cavities  we investigated the performance of lg in high finesse cavities by simulation with advanced ligo mirror maps .this illustrated the degraded purity of the circulating beam in realistic cavities due to higher order mode degeneracy .the results were then analysed by looking at the zernike polynomials representing our example mirror map .the analysis and results were consistent with the predictions made from eq .[ eq : k_eq ] .this suggested that astigmatism was causing a significant amount of coupling , particularly into the lg and lg modes .this was confirmed when the cavity was simulated again with the astigmatism removed from the mirror map and we observed a dramatic increase in the lg mode purity .the analytical description enabled us to identify the specific zernike polynomials which cause large couplings as well as the lg modes which would dominate as a result . using thiswe were able to derive certain requirements for our example mirror map , etm08 , in terms of limits on the amplitudes of the zernike polynomials z , z and z ( table [ table : limits ] ) . using this mapthe resulting circulating beam impurity was found to be 815ppm , a significant reduction from the original impurity of 0.114 .this investigation has demonstrated that a high beam purity is achievable using an lg beam when modifications are made to the low order zernike polynomials in advanced ligo mirrors . implementing the lg beam in gravitational wave detectorswill be challenging as we require very small amplitudes on these lower order polynomials .we should also consider that the example mirror surfaces considered here refer to uncoated substrates .the coating process is likely to add to the lower order features in the mirror surfaces .however , using this analytical approach we can derive specific requirements for the mirror surfaces leading to designs for suitable mirrors for these higher order beams .we would like to thank garilynn billingsley for providing the advanced ligo mirror surface maps and for advice and support on using them .we would also like to thank david shoemaker and stefan hild for useful discussions .this work has been supported by the science and technology facilities council and the european commission ( fp7 grant agreement 211743 ) .this document has been assigned the ligo laboratory document number ligo p1100081 .the product of two laguerre - gauss modes is : the following derivation follows from eq .[ eq : m_condition ] .currently we are concerned with the magnitude of the coupling coefficients , so we ignore any constant phase shifts and integrate with respect to : in order to further simplify the equation the following variable substitution is made : and a new limit to the integral : where is the zernike radius .this gives the integral : substituting in the sums representing the laguerre polynomials and the radial zernike function as a function of gives : this type of integral results in the lower incomplete gamma function : when is equal to , an integer , the function is given by the following sum : therefore , the final equation for this approximation of the magnitude of the coupling coefficients is given by : loss is given by : where the integral represents the normalised power reflected by a perfect mirror . defines an infinite plane perpendicular to the beam axis .the magnitude squared of an lg mode is : where is the beam radius , and are the mode indices and is the radial position . integrating over the surface and taking a similar approach as for the coupling coefficientswe get : where , is the radius of the mirror and is the lower incomplete gamma function .for certain zernike polynomials ( those with non - zero ) their amplitudes in a surface depend on the orientation of that surface with respect to the zernike surface .for example , consider the two polynomials responsible for astigmatism , z .the two polynomials actually describe the same shape , with one just rotated by 90 degrees with respect to the other . therefore , rotating a surface , such as the one described by mirror map etm08 , will change the amplitudes of these two polynomials within the surface . fig .[ fig : etm08_zs_rotated ] illustrates this effect .the plots show the amplitudes of the order 2 zernike polynomials present in the etm08 mirror surface as it is rotated .as expected the amplitude of the z polynomial remains constant as it has no angular dependence .the amplitudes of the z polynomials oscillate and , at a certain rotation ( around 120 ) the astigmatism of the surface is completely described by z , and 90 later completely described by z .the root mean squared amplitude ( ) of the polynomials remains constant .we have used zernike polynomials to describe mirror surface distortions and analyse the coupling that occurs from lg into other order 9 modes .this approach appears to be very suitable as we have been able to identify specific polynomials which extract significant amounts of power from the input mode .however , there is an alternative method used to investigate mirror surface distortions , which involves looking at the spatial frequencies present in real mirrors . in this sectionwe compare these two methods .to look at the spatial frequencies present in realistic mirrors we perform a 2d fourier transform of the surface height data .the resulting spectra is then analysed and synthetic maps are created with the same spatial frequencies .this method focuses on identifying particular spatial wavelengths which cause a large degree of coupling from lg .many synthetic maps are created and used in simulations of gravitational wave detectors . a statistical approachis then taken to determine the extent of the coupling when specific spatial frequencies are present in the mirror .  in the zernike approachwe look at the different polynomials present in mirror surface distortions .this can be thought of as equivalent to looking at the spectra of the mirror surfaces as the different polynomials represent different spatial frequencies .the plot in fig .[ fig : map_spectra ] illustrates this .the spectrum of the etm08 advanced ligo mirror map is shown , along with the spectra of maps made up from the zernike polynomials present in the etm08 mirror .each of the zernike maps recreates the ligo map with polynomials up to a certain order .the plots show that as the order of zernike polynomials present increases the higher spatial frequencies are represented in the mirror map .this is because these higher order polynomials represent the higher order spatial frequencies .looking at the spatial frequencies present in the zernike polynomials we found that the frequencies depended on the order , .a consequence of this is that if we just consider the spatial frequencies present in the mirror maps we will not be able to distinguish between polynomials with different . as we have seen , the azimuthal index is very significant as it determines which modes are largely coupled from lg . therefore looking at the spatial frequencies does nt identify the important shapes in the mirror surfaces .the zernike approach would seem to be the most suitable as this allows us to identify the interesting polynomials and modes .                `zernike polynomials and their use in describing the wavefront aberrations of the human eye ' ( stanford university , 2003 ) http://scien.stanford.edu/pages/labsite/2003/psych221/projects/03/pmaeda/index.html

Abstract: higher order laguerre  gauss ( lg ) beams have been proposed for use in future gravitational wave detectors , such as upgrades to the advanced ligo detectors and the einstein telescope , for their potential to reduce the effects of the thermal noise of the test masses . this paper details the theoretical analysis and simulation work carried out to investigate the behaviour of lg beams in realistic optical setups , in particular the coupling between different lg modes in a linear cavity . we present a new analytical approximation to compute the coupling between modes , using zernike polynomials to describe mirror surface distortions . we apply this method in a study of the behaviour of the lg mode within realistic arm cavities , using measured mirror surface maps from the advanced ligo project . we show mode distortions that can be expected to arise due to the degeneracy of higher order spatial modes within such cavities and relate this to the theoretical analysis . finally we identify the mirror distortions which cause significant coupling from the lg mode into other order 9 modes and derive requirements for the mirror surfaces .


recent development of information and communication technology empowers us both to collect and analyse a large amount of data on socio - economic systems .several researchers refer to the present epoch driven by this technological changes as _ the information explosion era _  .this progress enables us to monitor many details of financial assets and to record the information as a high frequency multivariate time series .recently , investigations of foreign exchange rates have been conducted by numerous researchers using various approaches based on statistical physics and time series analysis  .drod et al .showed that exchange rate return fluctuations for main currency pairs are well described by non - extensive statistics and possess multifractal characteristics  .rebitzky studied the influence of macroeconomic news on exchange rates  .he concluded the following three points : ( 1 ) fundamental news do matter , ( 2 ) news influences exchange rates by incorporating common information into prices directly and indirectly based upon order flow and ( 3 ) the impact of fundamental news on exchange rates is fairly stable over time .kaltwasser  estimated the herding tendency in the foreign exchange market for three currency pairs using the extended alfrano - lux model  and computed several unconditional moments of corresponding daily log - returns .gworek et al .analysed the exchange rate returns of 38 currencies ( including gold )  .they examined the cross - correlations between the returns of various currency pairs as well as between their signs , and in this way , they constructed a corresponding minimal spanning tree for several base currencies .liu et al . applied cross - sample entropy ( cross - sampen ) to compare two different time series in order to asses their degree of asynchrony to the daily log - return time series of foreign exchange rates in the currency markets  .they compared the correlations between time series with cross - sampen and showed that cross - sampen is superior in describing the correlations .moreover , theoretical models  and empirical analysis  on the tick - by - tick data of the foreign exchange market have been considered .ohnishi et al . proposed a weighted - moving - average analysis for the tick - by - tick data of yen - dollar exchange rates .they concluded that the weights decay exponentially with a time scale less than 2 min , implying that dealers are watching only the very recent market state .hashimoto and ito examined market impact of japanese macroeconomic statistic news within minutes of their announcements on the yen - dollar exchange rates  .they found clear increases in the number of deals and price volatility immediately after such announcements .  in principle , information arrivals can be never observed comprehensively ; however , one can register quotations(transactions ) from market participants .perceived information determines investor attitude , but the impact of news on trading decisions is an open question  .  nevertheless , we believe that it is important to confirm whether correlations exist between the number of quotations and transactions . in this study , we focus on the number of quotations or transactions for several currency pairs quoted in the foreign exchange market and we treat them as a multivariate time series that characterise system dynamics . bonanno et al . investigated the spectral density of both the logarithm of prices and the daily number of trades of a set of stocks traded in the new york stock exchangethey confirmed the behaviour of the spectral density of stocks log - returns and detected a -like behaviour of the spectral density of daily trade numbers .this shows that the number of transactions is generated by a long memory process , similar to the volatility dynamics .more conceptually , the mixture distribution hypothesis is one of the most famous explanatory models for price fluctuations in the financial market  .specifically , there is a positive simultaneous correlation between transaction volume , or the number of transactions , and returns  .furthermore , mandelbrot - taylor  and clark  proposed the concept of time changes or subordinated processes in order to explain the fat - tailedness of returns .it was reported that the number of trades is profoundly related to volatility .according to the normal variance mixture models in finance  , despite of the heteroskedacity of volatility with long memory , unconditional distribution of log - returns can be fitted to the mixture of normal distributions with an unconditional distribution of volatility .the same theory has been recently proposed in statistical physics by beck to investigate superstatistics  .therefore , it is important to capture the cross - sectional structure of quotations and transactions to understand their volatility .  in studies on financial time series , several researchers have taken the stance that traded volume , or the number of transactions ( quotations ) , is a proxy variable for unobservable information arrivals on market participants  .researchers of econophysics focus on several types of scaling relationships observed in the financial markets in order to understand the behaviour of market participants  .specifically , a scaling relationship between mean values of money flow and their standard deviation ( `` taylor s power law '' or `` fluctuation scaling '' ) is a useful scaling relationship when conducting multi - dimensional analysis on financial markets .a scaling relationship between a mean of constituents flows at the -th observation point and their standard deviation is referred to as fluctuation scaling : where represents a positive constant , and is a scaling exponent taking a value ranging from 1/2 to 1 .the pioneering work on fluctuation scaling is known as a taylor s study in ecology  .after his study , the scaling relationship between means and standard deviations has been found in a wide spectrum of fields .recently , menezes and barbasi have reported the scaling relationship on traffic flows , river flows and browsing activities of internet  .eisler and kertsz investigated fluctuation scaling of the traded value and dependence of the scaling exponent on a time scale  .more recently they computed the scaling relationship of the traded volume for 2647 stocks listed in the new york stock exchange ( nyse )  and 4039 listed in the nasdaq in the taq database , which records all transactions of the nyse and nasdaq during the period from 2000 to 2002 .they found a clear scaling relationship at these stock markets .they clarified that the hurst exponents and the scaling exponents depend on the window length .eisler et al . investigated fluctuation scaling on the volumes traded in the stock exchange market .furthermore , some researchers have examined cross - correlations among various quantities of the financial markets  .podobnik et al .investigated volume growth rate and volume changes for 14,981 daily records of the standard and poor s ( s&p ) 500 index over a 59-year period ( 1950 - 2009 )  . using detrended cross - correlation analysis , they found that there are power - law cross - correlations between them .bonanno et al .studied correlation - based network analysis of financial equities  .however , it is not obvious whether a stable scaling relationship exists in the foreign exchange market .this is the first focus of our paper .following the taylor methodology , we examine the scaling relationship between the mean of the numbers of quotations and transactions and their variance .we suppose that there should exist a relationship between fluctuation scaling for the market activity and cross - correlations at the foreign exchange market .this is the second focus of this article .this article is organised as follows . in sec .[ sec : empirical - analysis ] , we analyse the scaling exponents and cross - correlation matrices for a selected period . in sec . [sec : long - term ] , we conduct a long - term analysis of both the scaling exponents and the global average of the cross - correlations of both quotation and transaction numbers . sec .[ sec : conclusion ] is devoted to concluding remarks and used to address future works .in this section , we focus on the number of quotations arriving at the matching engine of the ebs platform and the number of transactions occurring there per unit time .we use comprehensive data ( icap ebs data mine level 1.0 ) collected by the ebs platform , one of the most popular electronic brokerage systems , which is exploited by over 2,000 traders at about 800 dealing rooms across the globe  .the data include records for orders ( bid / offer ) and transactions of currencies and precious metals during a period from june 2007 to december 2010 ( 43 months ) . during this period ,the global financial system suffered from the following significant macroeconomic shocks and crises : ( i ) paribas shock ( aug . 2007 ) , ( ii ) bear stearns shock ( feb . 2008 ) , ( iii ) sub - prime crisis driven by lehman shock ( sep .2008 to mar . 2009 ) and ( iv ) euro crisis ( apr . to may 2010 ) .let and be the number of quotations and transactions for exchange between the pairwise currencies during an observation period between and  , respectively .let us further assume that and are locally stationary variables in the time interval ].,title="fig:"](a )  and on the values of .( b ) scatter plots of the scaling exponents and the averaged correlation .the scaling exponents and the averaged correlation are computed from the quotation activities during a period from 8:38 , 3 august 2008 , to 21:56 , 8 august 2008 ( utc ) for the window length } \leq \delta t \leq 1500 \mbox{[min]})$].,title="fig:"](b )  moreover , we extend the global average of cross - correlation coefficients between the number of quotations and transactions . by using eq .( [ eq : cov - spd ] ) , we define the cross - correlation function matrix as and the global average of cross - correlations as obviously quantifies the anticipatory behaviour of the quotation activities against transaction activities .figure [ fig : pd ] shows the global average of the cross - correlations between the quotation activities and transaction activities .one can see that the global average of the cross - correlations possesses a maximum for the time difference .further , the positive tail of the cross - correlation function decays slower in time than does the negative one .now we show results for longer time periods .using icap ebs data mine level 1.0 , we performed the investigations of the scaling exponent and the global average of cross - correlations for both quotation and transaction activities . throughout this analysis , we fix [ min ] and [min ] . the empirical results are obtained from eqs .( [ eq : mean - sd ] ) and ( [ eq : cov - spd ] ) .we compute both the bootstrap mean of the scaling indices and their variance for each week .we generated 1,000 bootstrap samples with 100 points from observation points .we computed the temporal variation of scaling exponents for both quotations and transactions during all the observation weeks .it is found that the bootstrap standard deviation of the scaling indices is less than 0.01 .they vary in range from to .  in order to measure an error between empirical results and a fitting function with the least squars method, we define a norm of residual , where and ( ) are parameter estimates computed by using the least squared method .we show the norms of residuals and for the scaling relationship for each week in fig .[ fig : fs - residual ] .the figures show that the statistical quality of the scaling relationship between the mean values of activities and corresponding standard deviation , the error between the empirical relationships and eq .( [ eq : scaling2 ] ) with estimates , depends on the observation period .let us stress that if the norm of residuals is high then mean activities and their standard deviations may not be fitted to the scaling relation eq .( [ eq : scaling2 ] ) very well .specifically , the value of residuals calculated from the number of quotations suddenly increased or decreased before or after the periods of the ( i ) paribas shock ( aug . 2007 ) , ( ii ) bear stearns shock ( feb . 2008 ) , ( iii ) sub - prime crisis driven by lehman shock ( sep .2008 to mar .2009 ) and ( iv ) euro crisis ( apr . to may 2010 ) .the scaling relationship seems to break after the ( i ) paribas shock but before ( iii ) the sub - prime crisis and ( iv ) the euro crisis .the behaviour of residuals computed from the transaction activities is similar .however , the scaling relationship of the transaction activities is more stable than that of the quotation activities before or after the shocks .in addition , we compute the global average of the cross - correlation coefficients .the bootstrap standard deviations are approximately 0.1 during all observation periods .we found that the global average of simultaneous cross - correlations for quotation activities takes values less than 0.4 until february 2009 .from april to july 2009 , it took larger values than before .after july 2009 , the value increased and fluctuated around 0.5 , while the global average of transaction activities took smaller values .we conclude that the transactions occur in a mutually independent manner .  figure [ fig : alpha - corr ] shows scatter plots of the scaling exponent and the global average of simultaneous cross - correlation matrix .error bars show a 66.6% confidence level .the value of the correlation coefficient between and is 0.464 for quotation activities and that between and is 0.53 for transaction activities .there is both a relationship between and and between and .a solid line represents a fitted linear dependence obtained with the least squared method , where .we found that , for quotations and , for transactions .the rms of residuals , defined as where and are in the observation week , is 0.0734 for quotations ( ) and 0.0195 ( ) for transactions .it is confirmed that a relationship between the scaling exponent and the global average of cross - correlations is statistically significant .( p ) ( d )   and are 0.464 ( p ) and 0.53 ( d ) , respectively .the line represents .,title="fig:"](p )  and are 0.464 ( p ) and 0.53 ( d ) , respectively . the line represents .,title="fig:"](d )  moreover , using eq .( [ eq : average - pd ] ) , we compute the global average of cross - correlations between the quotation activities and transaction activities during the period from june 2007 to december 2010 .since the global average of their simultaneous cross - correlations always takes the maximum value , we compute them at .figure [ fig : ave - cross ] shows their global average for each week .the error - bars show a 66.6% confidence level obtained from the bootstrap standard deviations .the value of is not large but increases eventually .specifically , after august 2009 , the value rapidly increased .this significant increase coincides with scaling breaking of quotation activities . from this result, we conclude that in the ebs brokerage system , the quotation activities changed after august 2009 .we investigated quotation and transaction activities of the foreign exchange market with 43 months of high resolution data .we examined fluctuation scaling , the global averages of cross - correlation coefficients for both quotation and transaction activities and the global average of cross - correlation coefficients between them .we found that the scaling relationship between the mean of the numbers of activities and their variance is stable for a long observation period .however , we also observed that the scaling relationship breaks during several periods .specifically , the scaling breaking was observed from may to june 2009 .this may be related to imbalance between demand and supply of the foreign exchange market .the reason for this is an open question .we further found that there is a relationship between the exponents of fluctuation scaling and the global average of simultaneous cross - correlation coefficients .the relationship was found for both different window lengths and different observation periods .the scaling exponents have a positive association with the global average of cross - correlations .these values may measure the degree of the synchronous behaviour of participants .moreover , we recognised that the global average of cross - correlations between the quotation activities and transaction activities rapidly inclined after august 2009 .the occurred just after the significant scaling breaking of the quotation activities .the scaling breaking may be related to drastic changes in participants perception of the market .  in future research , the relationship between trends of exchange rates and scaling exponents could be explored .the link between the intuitive understanding of the prevailing market state by market participants and estimated scaling exponents should be examined .this work was supported by the grant - in - aid for young scientists ( b ) by japan society for the promotion of science ( jsps ) kakenhi ( # 23760074 ) ( a .- h .it was also financially supported by the japan securities scholarship foundation , grants - in - aid for scientific research ( a ) , ministry of education , culture sports , science and technology ( no .21243019 ) , and by ishii memorial securities research promotion foundation ( t. hayashi ) .this was also supported by the european cost action mp0801 physics of competition and conflicts and by the polish ministry of science grant 578/n - 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Abstract: we investigate quotation and transaction activities in the foreign exchange market for every week during the period of june 2007 to december 2010 . a scaling relationship between the mean values of number of quotations ( or number of transactions ) for various currency pairs and the corresponding standard deviations holds for a majority of the weeks . however , the scaling breaks in some time intervals , which is related to the emergence of market shocks . there is a monotonous relationship between values of scaling indices and global averages of currency pair cross - correlations when both quantities are observed for various window lengths .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore