ever since the first prediction of neutron stars ( ns ) by , ns have become a major test bed of various theories of dense matter and nuclear matter .the extreme high pressure and density achievable inside ns offers an ideal and unique environment to infer the equation of state ( eos ) of nuclear matter from astronomical observations . on the other hand ,the possible existence of quark stars ( qs ) would provide a strong support to a new form of matter , namely , the quark matter .moreover , the size and the mass of qs could also cast light on the properties of quark matter . thus , extracting useful information and constraints about nuclear and particle physics from astronomical observations of compact stars like ns and qs has become an active direction in astrophysics .since the density in the core of a ns ( or qs ) can be several times the normal nuclear density , the eos there is rather uncertain .various physical phenomena including pion condensation , hyperon formation and deconfined quark matter have been considered by theoretical nuclear physicists and sophisticated techniques of many - body calculations have been developed to yield a bunch of eos intended for the description of the deep interior of compact stars .it is standard practice for astrophysicists to construct compact stars with these eos for nuclear matter and establish various relationships among different physical quantities which could be obtained from astronomical observations .motivated by the possibilities of constraining eos of nuclear matter and extracting physical parameters of compact stars as well , researchers in astrophysics and nuclear physics have been actively seeking for eos - dependent and eos - independent relationships by examining the physical characteristics of compact stars constructed with different eos ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) .for example , compared the structure of compact stars of various kinds and discovered several empirical relationships connecting different physical characteristics of a star , such as the mass , the radius , the moment of inertia , and the mass distribution function as well .such relationships can be applied to infer the physical attributes of a compact star , including its eos , from astronomical observations .the main objective of the present paper is to unveil the universality embedded in the pulsation frequencies of non - rotating compact stars and to exploit these findings to infer the physical characteristics of ns or qs ( e.g. , mass , radius , moment of inertia and eos ) from their oscillation spectra .the pulsations of compact stars are an interesting and popular topic in its own right , for gravitational waves ( gw ) can be generated in such processes ( see * ? ? ?* for a seminal exposition of the topic ) .it is indisputable that the detection of gw would be a major milestone in general relativity and astrophysics .several earth - based gw interferometric detectors such as ligo , virgo , geo600 and tama300 have been operating .while the current detectors are still not sensitive enough to detect gw directly , interesting upper limits have been placed on either the gw strains or event rate for several potential astrophysical sources ( see , e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* for the results obtained from the latest science runs of ligo ) .owing to the energy carried away by gw , pulsations of compact stars are damped harmonic oscillations , which are analyzed in terms of quasi - normal modes ( qnm ) ( see , e.g. * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?each qnm is characterized by a complex eigenfrequency and has a time dependence , displaying exponential decay with a damping time .several attempts have been made to relate and ( or ) to the mass and the radius of a compact star and some universal relationships which are , to certain degree of accuracy , eos - independent have been found .however , most of these relationships fail to describe qnm of qs , which are stiff and self - bound .  on the other hand, found an empirical relation approximately expressing the moment of inertia in terms of the mass and the radius .such a relationship is universal as long as the eos under consideration is not too soft ( e.g. , hyperon matter ) or too stiff ( e.g. , quark matter ) .they proposed to use the universality to estimate the radius of a compact star from its mass and moment of inertia , with the latter two quantities being measurable for double pulsar systems by considering the spin - orbit coupling effect in general relativity .  motivated by the discovery of , we propose here to investigate the correspondence between the qnm frequency ( both and ) of the ( fundamental ) f - mode oscillations , and the moment of inertia of compact stars .we show that there exist eos - independent universal relationships between them , namely equations ( [ lau_r ] ) and ( [ lau_i ] ) , which are accurate up to a few percent or better and work nicely for qs as well . based on equations ( [ lau_r ] ) and ( [ lau_i ] ) ,a feasible scheme is in turn established to determine the mass and the moment of inertia of a compact star once its f - mode frequency is found from gw observations . as the radius of a ns ( or qs ) is approximately expressible in terms of the mass and the moment of inertia , our scheme can also yield good estimate of the radius and hence imposing constraints on the eos .the outline of the paper is as follows .section 2 is a brief review on the the behavior of f - mode oscillations of compact stars and the universal relationships among the mass , the radius and the moment of inertia of compact stars . in section 3we introduce an accurate universal relation between the scaled frequency and the physical quantity . in section 4we establish a feasible and robust scheme to apply our findings reported in section 3 to invert the mass , the moment of inertia , and in turn the radius of a compact star if its f - mode gw signal is detected .we finally summarize our paper in section 5 with a brief discussion .unless otherwise noted , we adopt geometric units with throughout the whole paper , and use kilometers as the unit of lengths .as mentioned in section 1 , the oscillation modes of compact stars are damped due to the emission of gw .hence , they are called quasi - normal modes , which can be further divided into polar and axial classes in each angular momentum sector according to the property of relevant metric and fluid perturbations under inversion . for typical non - rotating compact stars ,the polar class includes ( fundamental ) f - mode , ( gravity ) g - mode , and ( pressure ) p - mode and ( spacetime ) w - mode , while for non - rotating perfect fluid stars the axial class contains only the w - modes ( see , e.g. , * ? ? ?* for a review on the property of these modes ) .the methods for evaluation of qnm of compact stars are well documented in several papers and reviews ( see , e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?the study of qnm of compact stars has a long history dating back to the pioneering work of .it is by now well established that the study of gw mode spectra of compact stars could provide useful information about the internal structure of the stars ( see , e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?as it is expected that the direct detection of gw will be realized in the near future , different proposals have been put forward to determine the physical parameters ( e.g. , mass , radius and eos ) of a compact star from its gw spectra . in the present paperwe focus our attention on an eos - independent universal behavior of the quadrupolar ( angular momentum index ) f - mode and show that such universality can lead to accurate determination of the mass and the moment of inertia of a compact star ( ns or qs ) . combining our results with equations ( [ haensel_uni ] ) , ( [ lat ] ) and ( [ haensel_qs ] ) , one can also find an approximate value of the radius .the reason why we place special emphasis on the f - mode is that the relatively low frequency ( khz ) of the f - mode makes it more promising to be detectable by gw detectors than other modes .for example , considering a compact star 10 kpc from us , estimated that the energy required in the f - mode in order to lead to a detection with signal - to - noise ratio of 10 by the ligo detector is .for comparison , in order to achieve the same signal - to - noise ratio , the energy in the p - mode ( w - mode ) must be ( ) .the energy requirement is much relaxed for the advanced ligo detector and the same signal - to - noise ratio is achieved if the energy in the f - mode is . since the total energy released in violent astronomical events such as supernova explosions could be as much as in some optimistic estimates , it seems plausible that the f - mode qnm signal of a fiercely pulsating compact star inside our galaxy could be detected by future gw detectors , especially when the narrow - banding capability is available to enhance the sensitivity of the detectors in the high - frequency region at around 1 khz .there have been several attempts to search for an approximately eos - independent behavior of the f - mode . in particular, showed that the scaled frequency for the f - mode of realistic ns is essentially a universal function of the compactness , which can be approximated by the quadratic formulae : these two formulae are good approximation for most ns .however , the f - modes of qs do not follow equations ( [ leung1 ] ) and ( [ leung2 ] ) . instead , they are approximated better by another quadratic fit .the difference between the behaviors of the f - modes of ns and qs are expected because the density profile of a qs is nearly uniform , which is very different from that of a ns .  on the other hand , and discovered some empirical relationships relating and compactness . for ns, found that where the scaled moment of inertia and . also proposed an alternative expression : which yields better agreement for ns with higher compactness .however , both equations ( [ haensel_uni ] ) and ( [ lat ] ) fail to describe the case of qs . instead , found another relationship for qs : the distinction between the behaviors of ns and qs is also expected and attributable to the difference in the stiffness between nuclear and quark matters .  assuming that both and are obtained from observations, further argued that the radius can be found by solving these equation .however , two remarks about this method are in order .first , for the ns case the data points corresponding to stars characterized by different eos are rather scattered and may deviate appreciably from equations ( [ haensel_uni ] ) and ( [ lat ] ) . as a consequence, the value of obtained in this way is subjected to uncertainties depending on the eos .second , additional information has to be gathered to distinguish between the data of qs and ns beforehand .regardless of these remarks , it is observed that for a fixed eos the scaled moment of inertia increases monotonically with the compactness , indicating a one - one correspondence between these two quantities . in the following discussionwe shall consider the feasibility to replace the compactness used in equations ( [ leung1 ] ) and ( [ leung2 ] ) with the moment of inertia .it is interesting to note from the reviews sketched in sections 2 that both the scaled moment of inertia and scaled f - mode frequency are rather insensitive to eos and are , within certain accuracy , universal functions of compactness .such coincidence is interesting , reflecting the fact that compactness is a useful parameter characterizing a compact star .however , it is also noticed that the scaled f - mode frequencies ( or the scaled moments of inertia ) of ns and qs follow two different universal behaviors .the disparity in the universal behaviors of ns and qs is attributable to the difference in their density profiles .  in the present paperwe propose to examine the possibility of using the moment of inertia as an independent parameter , instead of the compactness , to characterize the f - mode spectra and in turn to search for any universality underlying these spectra .the physical motivation of our proposal is that the moment of inertia carries richer information about the mass distribution inside a star , which certainly affects f - mode oscillations .hence , the relationship between the moment of inertia and the f - mode is likely to be more direct and less eos - dependent .besides , as mentioned in section 2 , for compact stars with a fixed eos , the moment of inertia bears a one - one correspondence with the compactness . to this end, we replace the compactness by another dimensionless factor , which is proportional to the compactness for stars with uniform densities and is termed as the effective compactness .  the scaled f - mode frequencies for various realistic ns and qs are plotted against the effective compactness in figure [ fig : f_mode_real_i ] , where the f - mode qnm frequencies of the compact stars constructed with the following eos are shown :  .the solid lines in the lower and upper panels respectively represent the best quadratic fits in to and .in particular , the solid line in the lower panel is given by equation ( [ lau_r ] ) . ]  [ cols="^,^,^,^,^",options="header " , ]assuming that both and of the f - mode of a compact star have been obtained from gw observations , we can then solve the simultaneous equations ( [ lau_r ] ) and ( [ lau_i ] ) to determine the mass and the moment of inertia of the star .moreover , making use of equation ( [ lat ] ) proposed by , we can estimate the radius with the input of and .since all these three equations hold only approximately , the inferred values of , and are expected to deviate from the exact values by , , and . as shown in table [ uni_quality ] ,percentage errors in and predicted by equations ( [ lau_r ] ) and ( [ lau_i ] ) are small and usually less than a few percent .it is reasonable to expect that the percentage errors and are also small and such inversion scheme can lead to accurate determination of the mass and the moment of inertia of the star .  to validate the scheme outlined above and test its accuracy , we inferred , and numerically from and for three categories of compact stars , namely ns considered in figure [ fig : f_mode_real_i ] ( with au and apr1 eos ) , ns considered in ( with a , b and gm24 eos ) , and qs ( qs1 and qs2 ) .table [ invert_exp ] shows the percentage errors in mass , radius and moment of inertia of our scheme for these stars . in general , the inversion scheme worksvery well for ns and all three physical quantities can be determined up to a few percent or better .our scheme also reproduces accurate values of and for qs .however , the inferred value of is less satisfactory ( % ) .this is understandable as the empirical relation ( [ lat ] ) only works for ns , but not qs . if , however , there is other independent evidence ( e.g. , via the observed thermal x - ray emission ) suggesting that the star might be a qs , we could then use equation ( [ haensel_qs ] ) proposed by to infer a better value of from and .the percentage errors obtained in this way are enclosed by parentheses in table [ invert_exp ] .we see that the percentage error in is decreased from about 10% to 1% level . in summary, we expect that the inversion scheme proposed in the present paper is applicable regardless of the nature of the compact star in consideration .in this paper we study the universality embedded in f - mode pulsations of compact stars and propose to use it to determine the physical parameters ( including mass , radius and moment of inertia ) of a compact star from which the f - mode gravitational wave signal is detected .we first establish a pair of empirical equations ( [ lau_r ] ) and ( [ lau_i ] ) that can predict the frequency and the damping rate of f - modes from and with good accuracy for both ns and qs .we then apply such discovery to develop an inversion scheme to infer the mass , radius and moment of inertia of a compact star from its f - mode .in particular , the scheme is shown to work well for both ns and qs .the universality discovered in the present paper takes moment of inertia into the consideration , replacing the role of radius used in previous attempts .it is worth noting that the radius of a star is sensitive to the low - density part of its eos , whereas f - mode frequencies are expected to be dominated by dynamical behavior in the high - density regime .it is therefore physically appealing to use moment of inertia , which measures global mass distribution , to study f - mode oscillations . as clearly shown in table [ uni_quality ] , the replacement of with indeed leads to a much improved universal behavior . as it is arguable that the accuracy of the inversion scheme directly reflects the quality of the universal behavior upon which the scheme is based , the universality discovered in the present paper leads to an accurate inversion scheme .  finally , it is worthy of mentioning that the first gravitational - wave search sensitive to the f - modes has recently been carried out by the ligo detectors .while the current detectors have not yet detected gravitational waves directly , it is plausible that the advanced ligo detectors , which have a factor of 10 improvement on the sensitivity , might detect the f - mode gravitational wave signals from compact stars .the inversion scheme proposed in this work will provide a unifying way to infer the mass , radius and moment of inertia of both ns and qs accurately .of course in reality the situation is often complicated by the presence of magnetic field and rotation , in the future we shall consider the impacts of these factors on our proposal .this work is supported in part by the hong kong research grants council ( grant no : 401807 ) and the direct grant ( project i d : 2060330 ) from the chinese university of hong kong .we thank j. wu , y. j. zhang and p. o. chan for helpful discussions and assistance .

Abstract: we propose here a robust scheme to infer the physical parameters of compact stars from their f - mode gravitational wave signals . we first show that the frequency and the damping rate of f - mode oscillation of compact stars can be expressed in terms of universal functions of stellar mass and moment of inertia . by employing the universality in the f - mode one can then infer accurate values of the mass , the moment of inertia and the radius of a compact star . in addition , we demonstrate that our new scheme works well for both realistic neutron stars and quark stars , and hence provides a unifying way to infer the physical parameters of compact stars .   c i i


many complex phenomena can be represented as networks , and subsequently be analyzed fruitfully  .one of the first targets of network analysis is the detection of communities on the basis of the links , i.e. the possibly valued , or weighted , arcs or edges that connect the nodes .intuitively , an assignment of nodes to communities should be such that links within communities are relatively dense and between communities relatively sparse .this means we should compare actual densities to expected densities of links within and between communities .furthermore , since nodes , for example humans or proteins , can be members of different communities at the same time , e.g. organizations or protein complexes , respectively , the assignment should allow for the possibility that communities overlap .  in approaches to find appropriate community assignments ,much progress has been made in recent years  by using a concept known as _ modularity _  .while current modularity approaches take for granted that links are positively valued , representing bonds or attraction , scientists in numerous fields grapple with networks that also have _ negative _ links that represent repel , conflict , or opposition , for example in neural networks , semantic webs , genetic regulatory networks , and last but certainly not least , in social networks .  in this paper, we generalize an existing potts model  for positive links to incorporate negative links as well .we will follow the intuition that the assignment of nodes related by negative links should be done the opposite way of positive links , with negative links sparse within and more dense between communities , generalizing an old idea from social balance theory  .finally , we apply our approach to a network of conflicts and alliances between countries .recently , it was shown that modularity might miss small communities embedded in larger ones  , and is less accurate if the actual communities are highly different in size  .our method has two balancing parameters that address this problem to some extent  .yet community detection through modularity remains a global rather than a local approach .we consider a directed graph with nodes and links , which can be easily generalized to weighted graphs .we denote the total number of positive links in as and the number of negative links as , hence .we define the entries of the adjacency matrix of as follows : if a positive link is present from node to node , , if a negative link is present , , and otherwise . for a weighted graphthe link values , or weights , are denoted by .we separate the negative and positive links by setting if and zero otherwise , and if and zero otherwise , so .the positive and negative in- and outdegrees of are defined as   our challenge is to assign each node to one of communities . a complete configuration of community assignments is denoted by , which assigns each node to a community .the challenge of community detection in networks with positive and negative links was first addressed by social balance theory , which has its origins in cognitive dissonance theory  from the 1950s .this theory is based on the notion that if two people are positively related , their attitudes toward a third person should match .for example , if harry and mary are positively related as friends , and both of them are related to john , they should both be related to him either positively or negatively . in either casetheir triad is said to be socially balanced .if harry has a positive relationship with john while mary is negatively related to john or vice versa , their triad is socially unbalanced .if all triads in a network are balanced , the network is said to be balanced .this definition was later generalized to cycles , a triad ( a cycle of length 3 ) being a special case .the question whether a balanced network can be divided into separate parts arises naturally .the challenge is to define clusters of nodes such that there are only positive link within clusters and negative links are between clusters .it was proven  that if a connected network is balanced , it can be split into two opposing clusters ( and vice versa ) .however , there is an ambiguous case .if a triad has only negative relationships , it is neither balanced nor can it be split into two clusters .but it can obviously be split into three clusters . in order to accommodate for this possibility ,the definition of balance was generalized to -balance  .a network is -balanced if it can be divided into different clusters , each cluster having only positive links within itself , and negative links with other clusters .it can be proven  that a network is balanced if and only if it contains no cycles with exactly one negative link .the intuition is simple .suppose there is a cycle with one negative link , say between and , and only positive links between the remainder nodes , then and are both positively and negatively connected , and the cycle is unbalanced .but if in this cycle there is also a negative link between and , and , we can split the cycle in two parts , one cluster from to and one from to . if there are more than two negative links , we can split up the cycle analogously into more clustersin reality , however , social networks are rarely , if ever , fully -balanced .the question then becomes whether we can still cluster nodes . obviously , there are some links that make a network unbalanced. the number of such links can be expressed as an amount of _ frustration_. links that contribute to frustration are negative links within clusters and positive links between clusters .so the following expression should be minimized where if and zero otherwise , and is a parameter through which the contribution of the two types of links can be balanced .this is the approach taken in .the objective , then , is to find a clustering such that the frustration is minimized .since the term does not depend on the specific configuration and is therefore irrelevant for finding the minimum , we can simplify the above expression to we can now see that only for we retrieve , up to a multiplicative constant of . using any other value for would change the minimum found , but changing is the same as altering the ( weights of the ) original network . setting accordingly , we can simplify further , and now define frustration as   however , frustration does not generalize to a network with only positive links . in that case ,frustration groups together all nodes into one cluster .even if there are some negative links , frustration will cluster together very sparsely connected nodes .it s therefore clear that this approach does not match with current methods of community detection .preferably , there should not be a distinction between methods for positive and others for negative links , but there should be one method for both .in approaches to find appropriate community assignments in networks with only positive links , much progress has been made recently  .the principal method for detecting communities is through modularity optimization , which boils down to clustering nodes based on the link densities within and between communities .the link densities should be high within communities and low between communities .the ordinary , not , which is reflected in some minor changes to the original definitions  ; see also  . ]definition of modularity for directed graphs  is ,\ ] ] where is the actual number of links within a community , , and  ] is the expected number of arcs from to .hamiltonian ( [ equ : hamiltonian ] ) can be rewritten accordingly , where and are communities in .  in order to minimize the hamiltonian, we consider the effect of moving a single node from one community to another , expressed in terms of adhesion . more specifically , moving node from to results in the following change , where is the adhesion between node and its complement in community .let us write the mutual adhesion of a node and a community as .if , for the sake of argument , the mutual adhesion of and is larger than the mutual adhesion of and , decreases the hamiltonian .in other words , has more positive , or less negative , links than expected to than to , and moving to would improve the configuration .  to each movewe can assign a probability  where is the temperature and . by slowly decreasing the temperature , the probability of moving to another state approaches the maximum possible , thereby forcing the system into its minimum energy , i.e. the ground state .notice that in principle the probabilities are dependent on the total energy , but since we can simplify to the equation stated above .the algorithm iterates randomly over the nodes a number of times , after which the temperature is decreased to a lower temperature stepwise , and usually ( although not necessarily ) .the iterations and the lowering of the temperature are continued until there are no further ( significant ) improvements .any further changes would result in a higher energy , which we do not want , hence the resulting configuration of minimum energy is our solution .herein , for any set of nodes , its mutual adhesion to its own community is stronger than to any other community , which is clear when one looks at eq .( [ equ : change_hamiltonian ] ) .  furthermore , the cohesion , or self - adhesion , of a community is always positive , , and the mutual adhesion between two communities is always negative , .if the cohesion were negative , we could then move a set of nodes to another community and thereby decrease the energy , which would contradict the fact that the system is in the ground state .in fact these last two inequalities can be rephrased , which yields some insight into the effect of the parameters . if we assume , for analytic purposes , that , the expected values become = p^\pm n_rn_s ] , where is the number of nodes in community . writing this out we arrive at wherein the middle term is a sort of global density . hence by changing we change the threshold for clustering nodes together versus keeping them apart .either way , the density within a community is always higher than the global density of the network , while the density between communities is always lower than the global density .increasing raises the threshold for nodes to be clustered , and will ( generally ) result in smaller communities detected , possibly embedded in larger and sparser communities .increasing has the opposite effect and lowers the threshold , and will ( generally ) result in a configuration of larger communities .to show how our method can be applied to an empirical network , we analyze international relations taken from the correlates of war  data set over the period 19932001 , where military alliances can be represented by positive links and disputes by negative links .the data set contains a wide variety of disputes , for example border tensions between colombia and venezuela , the deployment of chinese submarines to japanese islands , and turkish groups entering iraqi territory .disputes were assigned hostility levels , from `` no militarized action '' to `` interstate war , '' and we chose the mean level of hostility between two countries over the given time interval as the weight of their negative link .the alliances we coded one of three values , for ( 1 ) entente , ( 2 ) non - aggression pact , or ( 3 ) defense pact .the disputes and alliances are both normalized to values in the interval $ ] which then bear equal weight in the overall link value .the largest connected component consists of 161 nodes ( countries ) and 2517 links ( conflicts and alliances ) .the result of the analysis ( ) is shown in fig .[ fig : map ] . countries of the same color ( or pattern ) belong to the same community , which in this context is more appropriately labeled a _ power bloc_. how strongly a country belongs to its power bloc can be determined by the adhesion a node has to its community .the power blocs can be identified as follows : ( 1 ) the west ; ( 2 ) latin america ; ( 3 ) muslim world ; ( 4 ) asia ; ( 5 ) west africa ; and , ( 6 ) central africa .if we detect communities by using only positive links , there is an agreement of about 64% with the configuration in fig .[ fig : map ] , while if using only negative links , there is an agreement of about 30% .our result resembles the configuration depicted in huntington s renowned book _ the clash of civilizations _ , with a few notable exceptions .the west african power bloc is an additional insight that is absent in huntington s configuration .a major difference with huntington is that china itself does not constitute a separate bloc , nor does japan or india .some other noteworthy differences are pakistan and iran which are grouped with the west , while south korea and south africa are grouped with the muslim world .if we run the algorithm with and , north america merges with latin america , while europe becomes an independent community , and north africa and the middle east align with russia and china .when setting and , in contrast , former soviet countries separate from russia and form an independent community . using a range of values for , one can detect various layers in the community structure .our configuration does not imply that conflicts take place between power blocs only , as 24% of all conflicts actually take place _ within _ blocs .for example ,georgia and russia had serious conflicts , and dr of congo and rwanda had theirs , but each of these pairs is grouped together nevertheless . in these cases , the alliances overcame the conflicts in the grouping , confirming that a configuration of international relations is more than the sum of bilateral links .  our political analysis hereis limited , since we wish to demonstrate the method rather than present a complete coverage of international alliances and disputes .other approaches that could be brought into play are the democratic peace theory  , which predicts few conflicts between democratic countries but fails to predict that in actuality , most conflicts occur between democratic and non - democratic countries ; the realist school  , which emphasizes geopolitical concerns ; and , the trade - conflict theory  , which argues that ( strong ) trade relations diminish the probability of a dispute , or lower its intensity . in sum , although huntington s configuration of civilizations was questioned  , it seems to be fairly robust and with some marked exceptions is confirmed by our analysis .we have extended the existing potts model by adapting the concept of modularity to detect communities in complex networks where both positive and negative links are present .this approach solves a long - standing problem in the theory of social balance , namely the clustering of signed graphs .  as a case in point , we have analyzed a social network of international disputes and alliances .other applications could be networks of references on the web  or in blogs  .if in these data positive and negative references are distinguished , our method makes possible to detect not only thematic clusters , but also positional clusters with internal agreement and external disagreement .  for network data , the model s parameters ( )can be used to find smaller ( sub ) communities , although there is currently no theoretical guidance to choose parameter values .even if there were such guidance , the modularity approach intrinsically aims at global rather than local optimization .our implementation is based on simulated annealing  , which performs quite well with standard tests , although for very large networks , faster algorithms will be necessary  .whatever algorithms future researchers will use , or improvements of the concept of modularity they will develop , being able to detect communities in networks with both positive and negative links is important in numerous fields of science , and a stepping stone toward further analyses of complex networks .  the authors like to thank anonymous referees and jean - charles delvenne for their valuable comments and advice .vt acknowledges support from a grant `` actions de recherche concertes large graphs and networks '' of the `` communaut franaise de belgique '' and from the belgian network dysco ( dynamical systems , control , and optimization ) , funded by the interuniversity attraction poles programme , initiated by the belgian state , science policy office .

Abstract: detecting communities in complex networks accurately is a prime challenge , preceding further analyses of network characteristics and dynamics . until now , community detection took into account only positively valued links , while many actual networks also feature negative links . we extend an existing potts model to incorporate negative links as well , resulting in a method similar to the clustering of signed graphs , as dealt with in social balance theory , but more general . to illustrate our method , we applied it to a network of international alliances and disputes . using data from 19932001 , it turns out that the world can be divided into six power blocs similar to huntington s civilizations , with some notable exceptions .


one of the fundamental problems in the foundations of statistical mechanics is to give an explanation as to why ` equilibrium ' statistical mechanics is so successful . that is to say , why the use of the standard gibbsian methods ` reproduces ' thermodynamic results .one offered explanation for this is the standard ergodic approach . as to whether this gives an acceptable explanation offers the impression `` that the _ communis opinio _ in the physics literature is that it does ; in the philosophy literature that it does nt '' .my impression agrees with hers .however , there is a further twist to the story . when confronted with the question of what is ` actually going on ' in a gas of particles ( say )when it is in equilibrium , or when it is coming to equilibrium , many physicists are quite prepared to desert the gibbsian approach entirely and to embrace a boltzmannian view . in particular according to ( ibid ) :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` ludwig boltzmann explained how irreversible macroscopic laws originate in the time - reversible laws of macroscopic physics .analysis is basically correct .the most famous criticisms of boltzmann s later work on the subject have little merit .most twentieth century innovations such as the identification of the state of a physical system with a probability distribution on phase space , of its thermodynamic entropy with the gibbs entropy of , and the invocation of the notions of ergodicity and mixing for the justification of statistical mechanics are thoroughly misguided '' ( p. 39 ) ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  and ( ibid ) :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` having results for typical microstates rather than averages is not just a mathematical nicety but at the heart of understanding the microscopic origin of observed macroscopic behaviour .we neither have nor do we need ensembles .what we do need and can expect is typical behaviour '' ( p. 38 ) ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  these assertions are reinforced by the opinion of ( ibid ) that the bolzmannian approach  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` is now generally accepted by physicists . there are some dissenting voices , such as that of ilya prigogine , but the disagreement is based on philosophical prejudice rather than physical evidence '' ( p. 113 ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  however , most work in equilibrium statistical mechanics uses the tools developed by gibbs . given a particular thermodynamic setup and microscopic model the appropriate probability distribution ( microcanonical , canonical , grand - canonical etc . ) is chosen .the entropy is taken to be that of gibbs and the holy grail for any investigation is an analytic form for the partition function ; the notable successes being the solution of the zero - field two - dimensional ising model by , of the six - vertex model in 1967 by lieb and of the eight - vertex model in 1972 by baxter ( see , * ? ? ?there have been many attempts to extend the gibbs approach to non - equilibrium . as indicated above in the quote from , the most developed programme for doing thisis that of the brussels austin school of the late count ilya prigogine .however , many remain unconvinced of either the actual or potential success of this enterprize. there would seem to be the need to explore the possibility of holding , with , , et al ., to the conviction that boltzmann got it right about the approach to equilibrium , whilst at the same time with a good conscience continuing to use the standard distributions of gibbs for everyday equilibrium calculations . we shall attempt to take some steps along this path . in order to doso we need to resolve in some way three questions , to which the current versions of the gibbs and boltzmann approaches offer apparently irreconcilable answers .a.  what is meant by equilibrium ?b.  what is statistical mechanical entropy ?c.  what is the object of study ?the attempt to produce conciliatory answers to ( i ) and ( ii ) will occupy most of this paper .however , we must at the outset deal with ( iii ) .ensembles are an intrinsic feature of the approach of for whom `` it is at the level of ensembles that temporal evolution can be predicted '' and for a `` thermodynamic system _ is _ [ my italics ] a system that has , at any given time , states distributed throughout phase space , and the distribution of these states is characterized by a density ''(p .the density referred to is the ensemble density and thus the ensemble becomes the way that a thermodynamic system is _defined_. in contrast to this we shall follow the view of given above that the object of study in statistical mechanics is a _single system_. all talk of ` ensembles ' is taken to be just a way of giving a relative frequency flavour to the probabilities of events occurring in that system .  in sec .[ micros ] we define the dynamic microstructure of our system and introduce two gas models which will be used as examples in the later parts of the paper . in sec .[ approaches ] we describe the different concepts of equilibrium in dynamics , the boltzmann and gibbs approaches to statistical mechanics and thermodynamics in sec .[ bmod ] we discuss the boltzmann approach with particular reference to the way in which equilibrium is defined and the neo - boltzmannian reliance on the concept of typical behaviour . with respect to the latter we suggest that the expected typical behaviour can be encapsulated by the term ` thermodynamic - like ' , which signifies the type of fluctuations to be expected in dependent thermodynamic variables , most particularly the entropy , and we discuss the conditions on the system needed for thermodynamic - like behaviour to be typical . at that pointit is necessary to consider the usual demarcation made between the system being or not being in equilibrium and we discuss , with the use of the example of the baker s gas with a small number of microsystems , the problematic nature of this property .this leads us to the rather radical suggestion that it would be better to replace the binary equilibrium property with a continuously changing property which we call _ commonness_. in sec .[ gibbs ] we use the example of a gas expanding in a box when a partition is removed to describe the role ascribed to the gibbs approach in the overall picture we are attempting to develop .of course , one question which needs to be addressed is the meaning of probability and this is discussed in sec .[ prop ] contains our proposals for a model incorporating both the boltzmann and gibbs viewpoints and our conclusions are presented in sec .consider a system that , at time , has a microstate given by the vector in the _ phase space _ .the time parameter can be discrete or continuous and the phase space can also be continuous or discrete . some one - to - one autonomous dynamics , ( ) determines a _ flow _ in and the set of points , parameterized by , gives a _trajectory_. the set of mappings is a semi - group .the system is _ reversible _ if there exists an idempotent operator on the points of , such that implies that .then and the set with or is a group .henceforth we shall assume that the system under discussion is reversible .if consists of a finite number of points then and the system is periodic . if is continuous then or . in this case , we shall restrict attention to a subset invariant under .we denote by a measure on such that ( a ) is finite , ( b ) is absolutely continuous with respect to the lebesque measure on , ( c ) is preserved by the flow ; that is , for all and all measurable .it can now be shown that the recurrence theorem will apply to flows for which .   to clarify the discussion it useful to resort to computer simulations of simple models. however , most interesting problems in statistical mechanics concern cooperative systems and , even at equilibrium ( see e.g. * ? ? ?* ) , there are few of these which can be solved exactly .so , of necessity , useful examples are of assemblies of non - interacting microsystems and the literature contains discussions of many ` toy models ' of this kind . herewe shall use two .   consider a gas of particles of unit mass moving in the two - dimensional box , , with elastic walls .we suppose that the initial positions , of the particles are such that all the are distinct and the initial velocities are all in the .then all the particles will perform one - dimensional oscillations in the at constant speeds .if we denote the positions and velocities at time by , then is constant and the system is reversible with . in fig .[ equ - fig1 ] we show a typical evolution of the gas from a state where all the particles are congregated near the left - hand end of the box .particles moving horizontally in a box with elastic walls.,width=340 ]  calculations for this example are used to illustrate the discussion in sec .[ prop ] .     particles moving under the bakers transformation.,width=340 ]this is the transformation , shown in fig .[ equ - fig2 ] , where a square of side is stretched to twice its width and then cut in half with the right - hand half used to restore the upper half of the unit square . as the mapping on the cartesian coordinates of the square , it is given by ( 2x,\ssfrac{1}{2}[y+1]),&{\rm mod}\ , 1 , & \ssfrac{1}{2}\le x\le 1 .\end{array}\right.\label{bak1}\ ] ] a convenient way of writing this transformation is to express and as binary strings : and with .then the transformation takes the form , with .the mapping is reversible with and .it can also be shown to be volume - preserving and thus that the recurrence theorem applies .we are interested in a baker s gas of ` particles ' . in fig .[ equ - fig3 ] we show a typical evolution from a state where all the particles are congregated near the bottom left corner of the box .the three levels of description in classical statistical mechanics , referred to in the abstract , each bring with them a definition of equilibrium .in fact by separating the macroscopic ( boltzmann ) and statistical ( gibbs ) approaches one can identify four distinct concepts of equilibrium .a * dynamic system * is in equilibrium at time if its state is given by a phase point lying on an attractor .however , of course , measure - preserving reversible dynamic systems do not have attractors . ''must not be taken too literally . ]  to describe the * boltzmann approach * we need to introduce some _macroscopic variables_. in general , these will be at the observational level for the system , but will encapsulate more detail than the thermodynamic variables .examples of these would be local variables which quantified spatial inhomogeneities of density or magnetization in a system ( see sec .[ bmod ] ) .tosses of a coin a microstate is a particular record of the outcomes and a macrostate is identified by the macrovariable and consists of the set of all microstates for which exactly of the outcomes are heads . ]suppose that , as in our examples discussed below , the system consists of identical microsystems and that we have a set of macrovariables .a macrostate is a subset of .the set of macrostates is defined so that :  a.  every is in exactly one macrostate denoted by .b.  each macrostate corresponds to a unique set of values for . c.  is invariant under all permutations of the microsystems .d.  the phase points and are in macrostates of the same size . is the same as that along .in fact , of course , in cases , like the perfect gas where macrostates are created by course - graining the one - microsystem configuration space and reversibility corresponds to changing the sign of the velocity and are in the same macrostate . ]given a particular set of macrovariables and macrostates , the _ boltzmann entropy _ is given by .\label{typ2}\ ] ] equilibrium is defined by reference to the macrostate , which is uniquely given by a set of values of the macrovariables . according to ,[briclab ] `` by far the largest volumes correspond to the _ equilibrium values _ of the macroscopic variables ( and this is how ` equilibrium ' should be defined ) '' .this is the standard boltzmannian definition .the system is in equilibrium if its phase point is in the macrostate of largest volume ( ) , corresponding to the largest value of , and this in turn defines the equilibrium values of the macrovariables .the starting point for the * gibbs approach * is to suppose the phase - point , in some invariant , is distributed according to a probability density function around a phase point is .the meaning we give to probability is discussed in more detail in sec .[ prob ] . ] which is invariant under the flow ( a solution of liouville s equation ) .equilibrium is defined as the situation where the probability density function is not an explicit function of time and becomes a function of the global constants of motion .the statistical mechanical ` analogues ' of thermodynamic quantities are either fixed external parameters , related to phase functions . ] or functionals of . in particular the analogue of thermodynamic entropy is the _ gibbs entropy _ =-k_{{{{\mbox{\tiny b}}}}}\int_{\lambda } \rho({{\boldsymbol{x}}})\ln[\rho({{\boldsymbol{x}}})]\,{{\mathrm{d}}}{{\eurscript}{m}}. \label{gibbs3}\ ] ] from a practical point of view this scheme is very satisfactory . however ,problems arise when an attempt is made to extend it to non - equilibrium situations , which are now perceived as being represented by time - dependent solutions of liouville s equation . specifically :  a.  when is replaced in ( [ gibbs3 ] ) by any time - dependent solution of liouville s equation ,  ] , where /(s_{{{\mbox{\tiny b}}}})_{{{\mbox{\tiny max}}}} ] into  ] , . ] and the atypical trajectories correspond to a choice of initial speeds which are commensurate and produce periodic , rather than quasi - periodic behaviour .however , ergodicity is not _necessary _ since ( [ prob2 ] ) is stronger than is required for a typical system to behave in a thermodynamic - like way and atypicality can also be broadened to include a set of trajectories of small but non - zero measure . as pointed out by ( , ) and verified by numerical calculations the kac ring model , which is not ergodic , exhibits clear thermodynamic - like behaviour .the typicality of thermodynamic - like behaviour , away from the perfect case of an ergodic system , is difficult to formulate .what counts as sufficiently close to the perfect situation of ( [ prob2 ] ) to yield an entropy profile which looks thermodynamic - like and how much atypical behaviour can be tolerated ? for the purpose of this work we shall leave these questions open with symbols of approximation standing as markers for a more detailed investigation .criteria described above for -equilibrium . ]ergodicity on is , of course , equivalent to being metrically indecomposable under the flow , and a non - ergodic situation where we can at least map out the elements of the discussion is when there is a finite or denumerable ergodic decomposition of .that is , to within subsets of zero measure , where each component is invariant and metrically indecomposable under the flow .ergodic decomposition will , of course , apply to any model ( like , for example , the kac ring ) where consists of a finite number of discrete points ; then each is a cycle of states .ergodic decomposition also applies where the system is -ergodic . in this caseone element of the decomposition ( say ) is such that for small .-ergodic , but as yet the hard evidence for this is lacking . ]  when ( [ tl1 ] ) applies , the time spent in the macrostate for all trajectories with ( apart from at set of measure zero ) is for such a system it is clear that temporal behaviour will be the same for all trajectories within a .the division must be between those members of the ergodic decomposition in which the behaviour is thermodynamic - like and those in which it is not .so :  1 .we indicate that the system is behaving in in a way which can be counted as thermodynamic - like by writing 2 .we denote by the union of all satisfying ( [ prob2b ] ) with and by the probability . ]that the system is in . then thermodynamic - like behaviour is typical for the system if that we are concerned to incorporate both boltzmannian and gibbsian view of statistical mechanics into our overall picture , we must now examine the role to be ascribed to the gibbs approach .this we do by means of a simple example .consider a gas of particles moving in dimensions .then the phase vector is -dimensional , composed of configuration and momentum vectors and , .let correspond to the gas being confined to the hypercubic box . then, if it is left undisturbed ` a sufficient time to attain thermodynamic equilibrium ' , the phase - point will , according to the gibbs prescription , be distributed in according to the appropriated equilibrium probability density function , which we denote by .now suppose that the gas is confined by a partition to the part of the box , denoted by , with , .in this situation , if the system is left to attain thermodynamic equilibrium , the appropriate probability density function will be , which differs from only in respect of the restriction on the configuration space .if the partition is removed at time the probability density function is 0,&\mbox{otherwise } , \end{array}\right . \label{gibbs5}\ ] ] where is that part of corresponding to all the particles being in .this is no longer the equilibrium distribution ; it will be the initial condition for a non - equilibrium solution evolving according to liouville s equation .however , as we have indicated , the gibbs entropy for this solution remains constant and does not converge to either in finite time , as we would like for thermodynamics , nor even on an infinite time scale .the most we can obtain is , for a mixing system , when the expectation values of a certain class of phase functions calculated using converge , as , to their expectation values calculated with .the resolution to this problem suggested by ( see also * ? ? ?* ) was to coarse - grain the phase - space in the manner in which macrostates were obtained in the boltzmann approach .in the macrostate the probability density function is replaced by where is the probability of the phase point being in . then from ( [ gibbs3 ] ) = -k_{{{\mbox{\tiny b}}}}\sum_{\{\mu\ } } \curpi(\mu)\ln[\curpi(\mu ) ] + k_{{{\mbox{\tiny b}}}}\sum_{\{\mu\ } } \curpi(\mu)\ln[{{\eurscript}{m}}(\mu ) ] .\label{extra21}\ ] ] the objections to this approach are well - known and we shall not discuss them here .we shall suggest a different approach .suppose that a phase point .what is the probability of the system being in a small measurable set around ?this will clearly depend on the physical circumstances of the system and will differ according to whether the phase point is confined to by the physical partition . in that case , if the system is in ` thermodynamic equilibrium ' the probability will be . however , if the partition is not present and the system is in ` thermodynamic equilibrium ' the probability will be .since corresponds to all the particles of the gas being in one end of the box , it will ( in our boltzmann language ) be a rather ` uncommon state ' and we expect that .now consider the case where at time the partition is removed .according to the gibbs prescription , the probability of is .the removal of the partition has not affected the probability . butcompare this situation with that where there has never been a partition present and the system phase point is in .what physically distinguishes the two cases ? if , in the course of its dynamic flow , the system is at at , its forward evolution will not affected by whether a partition has just been removed or whether it simply happens to have evolved into this state .so why should a different probability distribution be assigned to these two situations when ?will differ and the system from which the partition was removed can be regarded as suffering from ` false memory syndrome ' . ] a consistent approach , consonant with our treatment of the boltzmann approach , is to suppose that the only meaningful probability density function to be used from the gibbs approach is the time - independent solution of liouville s equation determined by the dynamics and the physical constaints on the system .a change of physical constraints will lead to an instant discontinuous change in the probability density function and the gibbs entropy .an uncommon state ( like , for example , the case of all the particles being in one end of the box ) will have low probability when calculated using and low boltzmann entropy , but the same gibbs entropy as any other configuration .in the gibbs approach of sec . [ gibbs ] we introduced the probability density function on the invariant set and in terms of this , in ( [ extra20 ] ) , defined the probability that the phase point will be in the macrostate .what we have not done is define what we mean by probability . within the gibbs approachthis is most frequently done using ensembles , which means that the probability density is not the property of a single system .but we have already indicated , in sec .[ intro ] , that our object of interest is a single system . our aim in this work is to bring some kind of reconciliation between the gibbs and boltzmann approaches and , as asserted by ( see the quotation in sec . [ intro ] ) , this latter neither has nor needs ensembles .we need two probabilities , that the phase point is moving on a particular trajectory and that on that trajectory the phase point lies in a particular subset of . for the latter we shall follow in using the _ time - average _ definition of probability .this analysis is simplified if , as we shall do , the ergodic decomposition ( [ tl1 ] ) is assumed . then :  a.  the probability was introduced in sec .[ typical ] .it can be taken to mean either the probability that when we choose the initial point it is in or the probability that when we investigate the system we find the phase point in . in either casethe probability is susceptible to a bayesian or laplacian interpretation .b.  the conditional probability that given that is , from ( [ prob2a ] ) , where the first equality represents the time - average definition of probability .then where denotes the expection value over the ergodic decomposition .  in each ergodic componentthere is a unique probability density function invariant under the flow given by 0,&\mbox{otherwise } \end{array}\right .\label{prob7b}\ ] ] and then , from ( [ extra20 ] ) , ( [ prob7b ] ) and ( [ prob7c ] ) , as given by ( [ prob7 ] ) and ( [ prob7a ] ) .  for any functions , integrable on , and , summable on the macrostates , the expectation value , of with respect to and with respect to ,are respectively clearly the time - average of along a trajectory in is and , from ( [ prob7 ] ) , ( [ prob7a ] ) and ( [ prob7ca ] ) , when the system behaviour in is thermodynamic - like it follows , from ( [ prob2 ] ) and ( [ prob2b ] ) , that a phase function which happens to be constant over each of the macrostates has , , giving . in particular , from ( [ prob7ca ] ) , . \label{prob5}\ ] ] relating the gibbs entropy to the boltzmann entropy will necessitate some rational choice for the probabilities .one view of is that it is the probability that , if we make a random choice for the the initial system point , then it lies in .if we assume that all points of are equally likely then on bayesian / laplacean grounds and consonant with the approach of we should choose then , from ( [ prob7b ] ) , ( [ prob7c ] ) and ( [ prob7ca ] ) which is the microcanonical distribution . from ( [ gibbs3 ] ) and ( [ typ2bis3 ] ) , \label{mer1}\ ] ] and , from ( [ prob5 ] ) , . \label{mer1b}\ ] ] when the system behaviour in is thermodynamic - like , it follows , from ( [ prob6b ] ) and ( [ prob6bb ] ) , that and , from ( [ prob2c ] ) , thermodynamic - like behaviour is typical if this section we shall make the following assumptions :  a.  that the system has the ergodic decomposition ( [ tl1 ] ) .. ] b.  that the probabilities for the system phase point being in members of the decomposition are given by ( [ assum1 ] ) . c.  that ( [ prob2cbis ] ) applies and thus that thermodynamic - like behaviour is typical.  assumption ( a ) is a purely dynamic assumption which may or may not be true for any particular system .the statistical assumption ( b ) is again based on the dynamics and has the same status as the assumption one might make about a die being unbiased when making a sequence of trials .assumption ( c ) has a dynamic aspect but is also dependent on some quantification of whether behaviour is or is not thermodynamic - like .  as indicated in sec .[ intro ] , two of the obstacles to overcome , in reconciling the boltzmann and gibbs approaches , are the different definitions of equilibrium and of entropy . with regard to equilibriumwe propose , as indicated in sec .[ ttde ] , that the two - valued condition of the system being or not being in equilibrium is replaced by a continuous property called _ commonness _ , based on a designation of macrostates related to a set of macrovariables .a microstate is more or less common according to whether the macrostate in which it is situated is of greater or smaller size .the boltzmann entropy is a measure of the size of macrostates and thus provides a measure of commonness .gibbs methods with a time - independent probability density function are to be retained as the practical means for obtaining an ` analogue ' of thermodynamics .the reason for using a time - independent probability density function to calculate thermodynamic properties is not that the system is in equilibrium but that the underlying dynamic is autonomous .the question of entropy can be subsumed into a more general account of the relationship between statistical mechanical and thermodynamic variables . as we have indicated in sec .[ approaches ] , the standard gibbs perception of the relationship between statistical mechanical and thermodynamic variables is that they fall into three classes .those for which  1 .the corresponding statistical mechanical variable is related to a phase function .the statistical mechanical variable and thermodynamic variable are identical and equal to an external parameter of the system .3 .  the corresponding statistical mechanical variable is a functional of the probability density function .the primary example of c3 is the entropy given by the gibbs formula ( [ gibbs3 ] ) .. ] in the interests of producing a smooth relationship between the boltzmann and gibbs approaches we shall make two modifications to the c1c3 scheme .the first , which is part of the standard boltzmann approach , is to related thermodynamic entropy to a phase function ( which is also a macrovariable ) , namely the boltzmann entropy ; so entropy is now of type c1 . the second is to propose a particular relationship , in the case of c1 variables , between thermodynamic variables and phase functions .  in sec. [ approaches ] we introduced the set of macrovariables defined on the macrostates .the scheme we now propose is that a thermodynamic variable , in class c1 , is related to a phase function , defined on , via a macroscopic function .the statistical mechanical variable is the macrovariable . in the case of entropythe phase function and macrovariable are identical and equal to the boltzmann entropy . for other thermodynamic quantitiesthis is not the case .we are , thus , proposing the three - part scheme , where we need to explicate the relationships denoted by ` ' .  1 .  :the value is the result of a measurement of , course - grained , to effectively give the same value throughout the macrostate .this perception together with the definition of macrostates and the boltzmann entropy in sec .[ approaches ] serves as the demarcation between the microscopic and macroscopic realms . of course , this demarcation is to some extent arbitrary , but it is equally so for any macroscopic physical theory . the value of the thermodynamic quantity , along the trajectory passing through , we take to be equal to the average of the result of a large number of measurements of taken at arbitrarily chosen times .so we can effectively define  is a constant of motion , but not constant on except when the system is ergodic .when has the ergodic decomposition ( [ tl1 ] ) , has the set of values from ( [ mer1a ] ) , when the behaviour in is thermodynamic - like , measurement of the thermodynamic variable will typically lead to a value close to .we now need to be more precise about what we mean by the macrovariable .ideally , of course , the identification in the notation of ( [ prob102 ] ) , would be desirable , because then it follows from ( [ merger2aa ] ) , that , for thermodynamic - like behaviour , this , to within the tolerance indicated by ` ' , is the standard identification between thermodynamic variables and the expectation values of phase functions in gibbs theory . as we have observed in sec. [ prob ] , ( [ merger3 ] ) is exactly true for the boltzmann entropy because it is , by definition , constant over each macrostate .we should not expect this to be true for all phase functions .but , of course , not all phase functions have a correspondence with thermodynamic variables .in particular we shall assume is ( a ) continuous on each macrostate and( b ) invariant under permutations of the microsystems .we have already assumed , in sec .[ approaches ] condition ( iii ) , that each macrostate is invariant under permutations of the microsystems .so the macrostate can be divided into non - overlapping subsets , each of identical .representing by a generic member of this set , it follows , from ( [ prob102 ] ) , that when macrostates are constructed from cells in the one - microsystem phase space , is the combinatorial factor in ( [ typ4 ] ) and .this latter becomes small as becomes large . in this case , since is continuous over , it is likely to vary very little . , given above , to the full set defining a finite - range observable ( , ibid ) .then , for large , the results of and could be used to assert that the dispersion of around is small . ]although the macrostate may be a very large part of it will be made of a large number of cells in each of which has the same nearly constant value .thus ' used here to indicated approximate equality , should be distinguished from ` ' which is used in this paper with the special sense of ` approximately equal in a system for which thermodynamic - like behaviour is typical ' . ] and   so to recapitulate the steps of the argument :  a.  thermodynamic variables are time - averages of phase functions course - grained ( to produces statistical mechanical variables ) over macrostates .b.  the thermodynamic variable is approximated along ( typical ) thermodynamic - like trajectories by the average of with respect to the probability density function .this establishes the relationship between a boltzmannian view of thermodynamic variables as time averages and the gibbsian view in which they are phase averages with respect to the appropriate probability density function .we now consider in more detail the case of entropy .it is often said that in `` equilibrium [ the gibbs entropy ] agrees with boltzmann and clausius entropies ( up to terms that are negligible when the number of particles is large ) and everything is fine '' . apart from the caveats we have entered in sec .[ ttde ] with respect to the concept of equilibrium , this assertion is based on two propositions ( i ) that the appropriate probability density function for the gibbs approach is given by the microcanonical distribution ( [ assum2 ] ) and ( ii ) that for large the ` equilibrium ' macrostate effectively fills the whole of .the first of these we have established , within our approach , for a system with an ergodic decomposition and with the choice of probabilities ( [ assum1 ] ) . in the case where macrostates are defined using the cell method described in sec .[ bmod ] , ( ii ) is not difficult to establish . using stirling s formula with ( [ extra3agis ] ) , }{n}\simeq\left\ { \begin{array}{ll } \displaystyle\frac{(p-1)\ln(n)}{n},&\mbox{with fixed ,}\\[0.5 cm ] ( 1+\xi)\ln(1+\xi)-\xi\ln(\xi ) , & \mbox{with , } \end{array}\right\}\hspace{0.15cm}\mbox{as . } \label{extra33}\ ] ] so in the case where , the number of cells is fixed , it follows from ( [ extra3gis ] ) that this is in agreement with the results shown in the diagonal figures in fig .[ equ - fig4 ] where we see that the value at which the entropy settles down is closer to the maximum entropy for larger values of . in the second case is the number of microsystems per cell . with both and large the convergence is effected by increasing the number of microsystems per cell ( )this is illustrated by the figures in the vertical columns of fig .[ equ - fig4 ] ., .the gas is allowed to evolved in the time interval confined to the left half of the box and then in the interval into the whole box .the upper and lower horizontal lines in each range correspond to and respectively . , width=377 ]  it also follows from ( [ mer1a ] ) , ( [ mer1b ] ) and ( [ extra3gis ] ) that as an illustration of these results we take the perfect gas and suppose that it is first confined by a partition to the region .the gibbs entropy , given by ( [ typ2bis3 ] ) and ( [ mer1 ] ) , is and for the respective cases where the gas is confined by the partition and where it is free to evolve over the whole box .the boltzmann entropy in the latter case is given by ( [ typ2 ] ) and ( [ typ4 ] ) , with and in the former case by the same formula but restricted by the condition that for .partitioned into strips . ]we consider the situation in which the system starts in a minimum entropy macrostate ( with all the particles in the first strip ) .it is allowed to evolve over the time interval  ] .this is shown in fig .[ equ - fig6 ] where in each range is well within the upper bound given by ( [ extra3gis ] ) and ( [ extra33 ] ) .in this programme we have used ergodicity and ergodic decomposition and as indicated in sec .[ intro ] there is deep ( and justified ) suspicion of the use of ergodic arguments to support the foundations of statistical mechanics , particularly among philosophers of physics . having given a comprehensive review of the problems of ergodic arguments , offerthe opinion `` that ergodic theory in its traditional form is unlikely to play more than a cameo role in whatever the final explanation of the success of equilibrium statistical mechanics turns out to be '' . in its ` traditional form 'the ergodic argument goes something like this : ( a ) measurement processes on thermodynamic systems take a long time compared to the time for microscopic processes in the system and thus can be effectively regarded as infinite time averages .( b ) in an ergodic system the infinite time average can be shown , for all but a set of measure zero , to be equal to the macrostate average with respect to an invariant normalized measure which is unique .the traditional objections to this argument are also well known : ( i ) measurements may be regarded as time averages , but they are not _ infinite _ time averages .( if they were one could not , by measurement , investigate a system not in equilibrium .in fact , traditional ergodic theory does not distinguish between systems in equilibrium and not in equilibrium . )( ii ) ergodic results are all to within sets of measure zero and one can not equate such sets with events with zero probability of occurrence .( iii ) rather few systems have been shown to be ergodic .so one must look for a reason for the success of equilibrium statistical mechanics for non - ergodic systems and when it is found it will make the ergodicity of ergodic systems irrelevant as well .our use of ergodicity differs substantially from that described above and it thus escapes wholly or partly the strictures applied to it . in respect of the question of equilibrium / non - equilibriumwe argue that the reason this does not arise in ergodic arguments is that equilibrium does not exist .the phase point of the system , in its passage along a trajectory , passes through common ( high entropy ) and uncommon ( low entropy ) macrostates _ and that is all_. so , although in our definition of a thermodynamic variable we have extended a large finite number of measurements to an infinite set of measurements , we can not be charged with ` blurring out ' the period when the system was not in equilibrium .the charge against ergodic arguments related to sets of measure zero is applicable only if one wants to argue that the procedure always works ; that is that non - thermodynamic - like behaviour never occurs .but we have , in this respect taken a boltzmann view .we need thermodynamic - like behaviour to be typical , but we admit the possibility of atypical behaviour occurring with small but not - vanishing probability .the way in which atypical behaviour can occur and demonstrates simulations of cases of this kind .] while the class of systems admitting a finite or denumerable ergodic decomposition is likely to be much larger than that of the purely ergodic systems , there remains the difficult question of determining the conditions under which the temporal behaviour along a trajectory , measured in terms of visiting - times in macrostates , approximates , in most members of the ergodic decomposition , to something recognizable as thermodynamic - like behaviour .i am grateful to jos uffink and dennis dieks for inviting me to make this contribution to the esf conference on philosophical and foundational issues in statistical physics , utrecht , november 2003 .i should also like to thank harvey brown , michael mackey , gerard watts and ivan wilde for useful discussions and janneke van lith and the referees for their constructive comments on an earlier version of the paper .bricmont , j. 2001 .bayes , boltzmann and bohm : probabilities in physics , in j. bricmont , d. drr , m. c. galvotti , g. ghirardi , f. petruccione n. zanghi ( eds ) , _ chance in physics : foundations and perspectives _ ( pp . 321 ) , springer - verlag .goldstein , s. 2001 .boltzmann s approach to statistical mechanics , in j. bricmont , d. drr , m. c. galvotti , g. ghirardi , f. petruccione n. zanghi ( eds ) , _ chance in physics : foundations and perspectives _ ( pp . 3954 ) , springer - verlag .lanford , iii , o. e. 1973 .entropy and equilibrium states in classical statistical mechanics , in j. ehlers , k. hepp h. a. weidenmller ( eds ) , _ statistical mechanics and mathematical problems _ ( pp . 1113 ) , springer - verlag .

Abstract: there are three levels of description in classical statistical mechanics , the microscopic / dynamic , the macroscopic / statistical and the thermodynamic . at one end there is a well - used concept of equilibrium in thermodynamics and at the other dynamic equilibrium does not exist in measure - preserving reversible dynamic systems . statistical mechanics attempts to situate equilibrium at the macroscopic level in the boltzmann approach and at the statistical level in the gibbs approach . the aim of this work is to propose a reconciliation between these approaches and to do so we need to reconsider the concept of equilibrium . our proposal is that the binary property of the system being or not being in equilibrium is replaced by a continuous property of _ commonness_. boltzmann ; gibbs ; approach to equilibrium ; ergodic decomposition .