the geometric nature of the thermodynamics has been considered as an important question since the pioneering works of gibbs and caratheodory . however , it was the development of differential geometry and einstein s theory of general relativity which increased the interest to extend and develop the geometric approach to other branches of physics . particularly , in thermodynamics ,the work of hermann and mrugala set up the manifold called _ thermodynamic phase space _, where the `` contact geometry '' approached by gibbs and caratheodory becomes meaningful .riemannian geometry was first introduced in the _ space of equilibrium states _ by rao , in 1945 , by means of a metric whose components in local coordinates coincide with fisher s information matrix . in fact , the fisher - rao metric can be considered as an element of the class of so - called hessian metrics whose local components coincide with the hessian of any arbitrarily chosen thermodynamic potential .rao s original work has been followed up and extended by a number of authors ( see , e.g. , for a review ) .the proposal coined by quevedo as geometrothermodynamics ( gtd ) was essential to unify both approaches and to endow the equilibrium states manifold with a legendre invariant metric .the importance of legendre invariance lies in the thermodynamics itself , meaning that once a representation is chosen to describe the system ( for instance , the internal energy or entropy ) , its legendre transform ( e.g. , the gibbs free energy or the massieu - planck potential ) contains the same information as the original representation .therefore , legendre invariance should be an essential ingredient of a geometric construction .basically , in the gtd approach , the thermodynamic phase space is endowed with a legendre invariant metric , and its maximally integral submanifold , that inherits its metric structure , is identified with the space of equilibrium states .in essence , a point of this space corresponds to an equilibrium state and , therefore , the thermodynamic processes take place in the equilibrium manifold . consequently, one expects the geometric properties of the equilibrium manifold to be related to the macroscopic physical properties .the details of this relation can be summarized in three points :  * the curvature of the equilibrium manifold reflects the thermodynamical interaction . *the phase transitions correspond to curvature singularities .* there exists a correspondence between quasi - static thermodynamic processes and certain geodesics of the equilibrium manifold .the geometrothermodynamic approach has been applied to classical systems such as the ideal gas and the van der waals gas , to more exotic systems such as black holes , and also in the context of relativistic cosmology to describe the evolution of our universe . in all the cases ,in which the analysis have been performed completely , the summarized items have been confirmed and the results have been shown to be legendre invariant .  in this paper , we aim to describe the geometry behind a chemical reaction .first , we will consider the case of a reaction with only two species , and then we will show that results can be generalized to include any arbitrary finite number of species .the paper is organized as follows . in section[ theory ] , we present a review of the geometrothermodynamic structures and the classical thermodynamics approach to chemical reactions ; particularly , we analyze the reaction in the context of classical thermodynamics . in sections [ sec : ig ] and [ sec :vdw ] , we consider the case of a reaction with two species corresponding to ideal gases and van der waals gases , respectively . in both cases , we present the thermodynamic analysis of the reaction to find the final state of equilibrium , and analyze the same situation from the point of view of gtd .it is shown that each reaction can be represented by a geodesic in the equilibrium manifold .then , in section [ sec : gtdcr ] , we show the applicability of gtd to a general chemical reaction .finally , section [ sec : con ] is devoted to the conclusions .the idea behind the geometrization of a thermodynamic system is simple : to build a space where each point corresponds to an equilibrium state .the physics behind the equilibrium thermodynamics allows us to say that this space is an -dimensional manifold with the dimension corresponding to the number of thermodynamic degrees of freedom of the system . in consequence, one needs only independent variables to coordinatize the manifold .  in standard equilibrium thermodynamics , to a system with degrees of freedomit is possible to associate extensive variables , intensive variables , where the index runs from 1 to , and a thermodynamic potential , relating them . in this context , the terms extensive and intensive are general concepts that refer to the independence or dependency of the variables associated to a given potential . for example , for a closed simple system with two degrees of freedom , the independent variables are and , if the potential chosen to describe the system is , or and , if the fundamental potential is .recall that .consequently , from the point of view of thermodynamics , to a system with degrees of freedom we associate variables , of them being independent .geometrically , this idea corresponds to an embedding of an -dimensional manifold into a -dimensional manifold given by [ gtd.emb ] : , or , in coordinates , : \{e^a } \{(e^a),i^b(e^a),e^a } , where also goes from to .the manifold is a contact manifold .this means that is endowed with a family of tangent hyperplanes ( contact structure ) defined by the so - called fundamental 1-form that satisfies the non - integrability condition [ gtd.ni ] ( ) ^n 0 .the manifold is called the _ thermoynamic phase space_. it can be coordinatized by the variables , where .its existence is relevant because it results adequate to perform legendre transformations ( as in thermodynamics ) as a change of coordinates .formally , a legendre transformation is a contact transformation , i.e. , a transformation which leaves the contact structure unchanged ; in coordinates , it is defined as  = - _ kl ^k^l , e^i = -^i , e^j = ^j ,  i^i = ^i , i^j = ^j , where is any disjoint decomposition of the set of indices , and .according to the darboux theorem , the 1-form of equation can be given in the coordinates as : [ gtd.local ] = - i_a e^a where we use einstein s summation convention for repeated indices .it can easily be seen that after a legendre transformation , the new 1-form in coordinates reads [ gtdlegendre ] = - _ a ^a .this proves that the contact structure remains unchanged .on other hand , the manifold is the maximally integral submanifold of , and is defined in such a way that the properties of the thermodynamic systems are encoded in it .so far , we have seen that is specified through the embedding , which is equivalent to specifying the fundamental equation of the system .the next step is to introduce the relations of standard equilibrium thermodynamics into the definition of the manifold . to this end, we demand that the embedding satisfies the condition [ gtd.01 ] ^ * ( ) = 0 , where is the pullback of . in coordinates, it takes the form [ gtd.02 ] ^ * ( ) = ^*(- i_a e^a ) = ( - i_a)e^a = 0 .it follows immediately that [ gtd.03 ] = ( e^a ) = i_a .equations and constitute the standard gibbs relations of equilibrium thermodynamics in , namely , = i_a e^a .the manifold defined in this way is called the _ space of equilibrium states_.  in addition to the geometric description of thermodynamics in terms of a contact structure , the gtd program promotes the contact manifold into a riemannian contact manifold , where is a metric sharing the symmetries of .the most general metric invariant under total and partial legendre transformations that has been found so far is given by [ quevedogiii ] g = + ( z^a ) _a=1^n , where is an arbitrary legendre invariant function of the coordinates and is an integer .the corresponding induced metric in the space of equilibrium states is given by \ .\ ] ] as suggested in , this metric could be useful to analyze multicomponent systems , particularly systems where chemical reactions take place .we will show in the next sections that , in fact , chemical reactions can be represented as geodesics of the equilibrium manifold described by the metric ( [ quevedogii ] ) .notice that to compute the explicit components of this metric , it is necessary to specify only the fundamental equation .thus , all the geometric properties of the equilibrium space are determined by the fundamental equation .this is similar to the situation in classical thermodynamics where the fundamental equation is used to determine all the equations of state and thermodynamic properties of the system .notice that the metric contains the arbitrary parameter which , however , can be absorbed by renaming the coordinates as and . then , [ giiixy ] g = + ( z^a ) _ a=1^n ( x^a y_a ) .furthermore , the arbitrary function can be fixed by demanding invariance with respect to changes of representation , an issue which is outside of the scope of the present work .it is therefore possible to perform the entire analysis with an arbitrary function in coordinates ; however , the physical interpretation of these coordinates becomes cumbersome and makes it difficult the physical interpretation of the results . for the sake of simplicity, we will use in this work the particular choice and , which has been shown to be useful also to describe geometrically systems like the ideal gas or van der waals gas . then , in the particular case , the metric of the equilibrium manifold reduces to \frac{\partial^2 \phi}{\partial e^1\partial e^2 } d e^1  \otimes d e^2 \ . \label{metrica}\end{aligned}\ ] ]  to analyze the geometric properties of the equilibrium manifold , we will consider the connection and the curvature .in particular , the connection is used to represent the geodesic equations in a given coordinate system + ^a _ bc = 0 , ^a _ bc = g^ad ( + - ) where are the christoffel symbols .the solutions of these equations are the geodesics , where is an affine parameter along the trajectory .one of the main goals of the present work is to show that a given chemical reaction can be represented geometrically as a family of geodesics of the equilibrium manifold , which is determined by the fundamental equation of the chemical system .the curvature tensor is defined as [ curvature ] r^a _ bcd = - + ^a _ e c^e _ bd - ^a _ e d^e _ bc . in gtd , the curvature tensor of the equilibrium manifold is expected to be a measure of the interaction between the components of the thermodynamic system .furthermore , from the curvature tensor one can define the ricci tensor and the curvature scalar .notice that in the case of a two - dimensional space , the curvature tensor has only one independent component , say and , therefore , the ricci tensor and the curvature scalar are proportional to .consider the general chemical reaction , [ cr ] in which the are the stoichiometric numbers of the reactants , and are the stoichiometric coefficients of the products , respectively .notice that the species need not all to occur in the same phase .the main condition for the chemical - reaction equilibrium in a closed system is that [ equilib ] _ i _ i_i = 0 , where the coefficients refer to the stoichiometric numbers and represents the chemical potential of the .notice that the condition ( [ equilib ] ) holds no matter how the closed system reaches its final equilibrium state . during a reaction , the change of numbers of moles of the species , , where is the number of moles of species at the beginning of the reaction , is proportional to the stoichiometric number with the extent of reaction as the proportionality factor , . for an infinitesimal extent of reaction , it holds that .the fact that the extent of reaction can be treated as an infinitesimal quantity is essential for the geometric description we will present below .the main premise to apply classical thermodynamics in closed systems , where chemical reactions can occur , is that we can use thermodynamic variables to describe the system even if it is not in material equilibrium .it means that variables such as , , , , etc ., are completely defined at any extent of reaction .  though is useful for practical situations, it does not contain information about the behavior of the different thermodynamic potentials from initial conditions to equilibrium . to obtain this information, we will use the fundamental equation of the chemical system .let represents the fundamental equation of the species .then , the fundamental equation for the general reaction ( [ cr ] ) can be constructed as follows notice that in this expression we are taking into account all possible interactions between all the species .the only assumption is that the interaction between the species and depends on the variables and , only .it seems that this condition is not very restrictive in realistic situations .  in the last subsection, we emphasized the role of the legendre transformations from the geometrical point of view .now , we will mention their importance from the thermodynamic point of view .  1 .  it does not matter which potential is chosen to describe a particular system ,all of them will contain the same thermodynamic information .the prediction of the final equilibrium state is made in accordance with the extremum principles " ; these principles could be different for different potentials .the importance of the legendre transformations is that they always can be used to find a potential in which the extremum principles hold for the experimental working conditions .  for later use ,we summarize here the `` extremum principles '' :  * _ entropy maximum principle ._ the equilibrium value of any internal unconstrained parameter is such as to maximize the entropy for the given value of the total internal energy . * _ energy minimum principle . _the equilibrium value of any internal unconstrained parameter is such as to minimize the energy for the given value of the total entropy .* _ general minimum principle for the legendre transforms of the energy . _the equilibrium value of any unconstrained internal parameter is such as to minimize the legendre transform of the internal energy for a constant value of the transformed variable(s ) . * _ general maximum principle for the legendre transforms of the entropy . _the equilibrium value of any unconstrained internal parameter is such as to maximize the legendre transform of the entropy for a constant value of the transformed variable(s ) .consequently , we can talk about legendre invariance in two senses .firstly , in the sense that the thermodynamic information remains conserved under a legendre transformation and , secondly , in the sense that the same equilibrium value for one or several internal unconstrained parameters will be obtained , independently of the thermodynamic potential , under the condition that the experimental restrictions are according to the particular restrictions contained in the `` extremum principle '' for .  in the following sections , we will study the chemical reaction considering and either as ideal monoatomic gases or as van der waals monoatomic gases .if we consider the species as ideal gases , the corresponding fundamental equation for each species in the entropy representation reads : [ ecfunds ] s(u , v , n ) = n s_0 + nr , where , , and refer to the standard values of reference , is a dimensionless constant related to the heat capacity of the ideal gas , i.e. , , and is the universal gas constant .let us consider the particular case in which only two species take part in the reaction .it turns out that it is convenient to study the evolution of the total entropy in terms of the extent of reaction . according to eq ., the fundamental equation takes the form \notag  \\  & + & ( n_{b,0}+\xi ) s_{0,b } + ( n_{b,0}+\xi ) r \ln\left[\left(\frac{u_b}{u_{0,b}}\right)^{c_b}\left(\frac{v_b}{v_{0,b}}\right ) \left(\frac{n_{b,0}+\xi}{n_{0,b}}\right)^{-(c_b+1 ) } \right ] \ , \label{fundamental}\end{aligned}\ ] ] where we neglected the interaction term for simplicity , and introduced explicitly the variable extent of reaction . note that refers to the initial conditions and refers to the values of the state of reference . for simplicity the values for and set equal to one .moreover , the values for and -which depend on the nature of each gas - are chosen in such a way that they basically take into account the differences between the species .these and other values are shown in table [ condiciones ] ..conditions for the gases a and b [ cols="<,^,^",options="header " , ]   finally , we take the temperature as and the volume as , experimental conditions that can easily be achieved . with these conditions, the fundamental equation reduces to s ( ) = 1++ r ( 1- ) ( ) + r ( ) , [ trabajo ] where the value of the constants have been rounded to simplify the presentation .a plot of this function is displayed in figure [ compa1 ] .since the temperature is constant and , the total internal energy is constant .thus , according to the entropy maximum principle " , we have that the maximum of , as a function of the extent of reaction , corresponds to the equilibrium condition . in this case, reaches it maximum value at .  to illustrate the significance of legendre invariance, we can analyze other potentials that are obtained from by means of legendre transformations .consider , for instance , the massieu potential ( helmholtz free energy ) considering the assumption with no interacting term , we obtain where . to calculate explicitly this function we use in this way , the fundamental equation of the systemis : - cr  \right\}\ . \label{massig}\end{aligned}\ ] ]  taking into account the considerations discussed for the entropy representation , and the value , finally the fundamental equation in this representation is reduced to ( ) = -11.47 + + r ( 1- ) ( ) + r( ) , where again we have rounded the values of the constants for simplicity .the plot of this function is displayed in figure [ compa2 ] .0.3   0.3    in this case , the transformed variable of the legendre transform is , and since it is a constant , the maximum of corresponds to the equilibrium state , which is reached at .thus , in agreement with the legendre invariance , we obtain the same result as in the entropy representation .the main result of this thermodynamic analysis is that a system with initial parameters as given in table [ condiciones ] , which correspond to an entropy , undergoes a chemical reaction whose final state is the equilibrium state characterized by the reaction extent and the entropy .notice that for ( ) all the states with ( ) are not permitted .in fact , once the system reaches the final equilibrium state at , the chemical reaction ends , and the states characterized by a decrease of entropy are unphysical according to the second law of the thermodynamics .   recall that to construct the metric of the equilibrium manifold we only need the fundamental equation . under the restrictions corresponding to the chemical reaction of two ideal gases as described in the last subsection , the original fundamental equation ( [ fundamental ] ) reduces to a function that depends on two variables only , namely , where we used the variable to rewrite the variables and as u_a = u , u_b = u . in the entropy representation , we choose the independent variables as .then , the metric of the equilibrium space ( [ metrica ] ) for the chemical reaction of two ideal gases reduces to g_s^ig = + .[ gig ] for the metric ( [ gig ] ) the only non - vanishing christoffel symbols are and .then , the geodesic equations read u ( ) - ( u ( ) ) ^2 = 0 , [ ec1 ] ( ) + ( ( ) ) ^2 = 0 .[ ec2 ]  the main point now is to see whether the geodesic equations can reproduce the thermodynamic process that occur during a chemical reaction .the idea is that we use the initial values of the thermodynamic variables , corresponding to the initial equilibrium state of the chemical reaction , to identify a particular point in the equilibrium manifold .this point is then used as initial value to integrate the geodesic equations .the question is whether the solution of the geodesic equations passes through the final equilibrium state of the chemical reaction .  for the particular case of ideal gases we are investigating here, we found in the last subsection that the thermodynamic analysis establishes the value of the extent of reaction for the final state .the values of the initial state have been incorporated in the fundamental equation ( [ feqig ] ) and , consequently , in the thermodynamic metric ( [ gig ] ) and in the geodesic equations .we now consider the experimental " condition .then , we get const ., so that equation is satisfied identically .then , we proceed to solve numerically the remaining geodesic equation for .the results are displayed in figures [ ideal1 ] and [ ideal2 ] .we choose as initial conditions values very close to and ( since the reaction can go in both directions ) , and different arbitrary initial `` velocities '' .the important result is that all the geodesics reach the point , independently of the initial values of and , and all of them follow a pattern in the physical region between the initial value and , where the entropy increases .in fact , the numerical integrator finds always a singularity " at the point .we will see below that in fact this point corresponds to a coordinate singularity of the thermodynamic metric .we now test the legendre invariance of our analysis by using the massieu potential ( [ massig ] ) .we insert the conditions for the present chemical reaction and obtain the corresponding fundamental equation . in this case, and the metric ( [ metrica ] ) leads to g_^ig= - .the only non - vanishing christoffel symbols are and .then , the geodesic equations are ( ) - ( ( ) ) ^2 = 0 , [ ec3 ] ( ) - ( ( ) ) ^2 = 0 .[ ec4 ]  we now fix the `` experimental '' condition , with that the equation is satisfied identically and we proceed to solve numerically the remaining equation ( [ ec4 ] ) for .the initial conditions are the same as in the entropy representation .the results are displayed in figures [ ideal3 ] and [ ideal4 ] .we see again that all the geodesics pass through the point , independently of the initial values .moreover , in the physical region , before the final equilibrium state is reached , they all coincide with the geodesics shown in figs .[ ideal1 ] and [ ideal2 ] for the analysis in the entropy representation .this shows explicitly that the analysis does not depend on the choice of thermodynamic potential .  0.4    0.4    0.4    0.4    a straightforward computation shows that the curvature tensor vanishes in both representations and , consequently, the corresponding equilibrium space is flat .this indicates that no thermodynamic interaction exists in a chemical reaction in which only non - interacting ideal gases are involved .a more realistic gas is described by the van der waals fundamental equation s(u , v , n ) = n s_0 + nr , where and are constants . for simplicity , we will consider both gases with the same value , so that the coupling terms in the fundamental equation can be considered as vanishing .that is , this case is a simple mixture in which the interactions are identical to the interactions and .consequently , the fundamental equation of the mixture reads [ mas ] s(u_a , u_b , v , n_a , n_b ) = s_a(u_a , v , n_a ) + s_b(u_b , v , n_b ) .  in the previous section ,the constant temperature condition of the ideal gas reaction was equivalent to implying . and , therefore , it was straightforward to compare the and representations .in fact , the conditions , under which the extremum principle is valid , were fulfilled in both representations . on the other hand , in the case of van der waals gases , the constant temperature condition does not imply that the total internal energy is constant .consequently , it is necessary to be cautious when applying the extremum principle corresponding to a constant temperature condition . to this end, we perform a legendre transformation in the to obtain the massieu potential so that the temperature is an independent variable .then , for the chemical reaction of two van der waals gases we obtain the thermodynamic potential \ . \label{feqvdw}\end{aligned}\ ] ]  for the sake of simplicity , we consider the same initial values as given in table [ condiciones ] for ideal gases , with the additional values and , which are quite reasonable for real gases . the behavior of the potential is displayed in fig .[ vdwmas ] .   considering and as van der waals gases . ]  notice that in this case the maximum is reached at which is only slightly different from the ideal case , because the initial equilibrium state is the same in both cases and , at k , the differences between ideal and van der waals gases are very small . in the next section , we will find out if there exists a geodesic in the equilibrium space that connects the initial equilibrium state with the final one .   considering the initial conditions given in table [ condiciones ] , the fundamental equation in this casecan be obtained directly from eq.([feqvdw ] ) as from here one can compute all the components of the metric tensor .the explicit expressions are quite cumbersome and can not be written in a compact form .the curvature scalar is in general different from zero , indicating the presence of thermodynamic interaction .this is in accordance with the statistical approach to the van der waals gas in which , as a result of the interaction between the particles of the gas , the corresponding hamiltonian possesses a non - trivial potential term .all the christoffel symbols are different from zero so that the geodesic equations contain all the independent terms this system is solved numerically with initial values for very close to and and arbitrary initial `` velocities '' . moreover ,for we introduce the `` experimental '' condition in the form and .the results are displayed in figs .[ vdw1 ] and [ vdw2 ] .we see also in this case that all the geodesics reach their final equilibrium state for which is the same value we obtained in the thermodynamic analysis of the last subsection .moreover , the geodesics follow a pattern in the physical region , in the same way as in the case of ideal gases .we conclude that the geodesics can be interpreted also in this case as the geometric path that the chemical reaction follows until it reaches its final equilibrium state .  0.4    0.4in the previous sections , we analyzed two particular chemical reactions with only two species . in both cases, we found that the corresponding equilibrium manifold reduces to a manifold , once the conditions of the reaction are taken into account .the question arises whether the dimension of increases as the number of reactants and products of the reaction increases .we will show in this section that gtd can handle in a simple manner the chemical reaction of any arbitrary ( finite ) number of species in a closed system .  consider the chemical reaction of species described by the variables , , , and  . in the entropy representation , for instance, the thermodynamic properties of each substance is determined by the fundamental equation . each fundamental equation generates a equilibrium space for the . according to eq.([total ] ) , the fundamental equation of the chemical system will depend on variables .then , the total equilibrium space has dimensions . in general , the level of computational difficulty in geometry increases with the number of dimensions , so that for large the calculations could easily be outside the reach of computational capability .however , we can use the conditions of the reaction to reduce the number of dimensions .a common condition for reactions involving gases is that the reaction occurs at constant volume so that the functional dependence of the fundamental equation can be reduce to , i.e. , .furthermore , using the definition of the extent of reaction parameter , , we can replace all the s variables by , according to .since and are constants , the functional dependence of the fundamental equation reduces to . using the equations of state for each species, we can express each in terms of , and other constants .for instance , in the case of ideal gases we have that u= _i u_i = _ i c_i n_i r t = rt _ i c_i ( n_i,0 + _ i ) .then , we can express each as u_i(u , ) = u , so that the fundamental equation becomes .consequently , the corresponding equilibrium manifold is 2-dimensional , independently of the number of species .  in the case of more complicated fundamental equations, it is always possible to express each in terms ofu and the extent of reaction in such a way that the resulting equilibrium manifold has only two dimensions .for instance , in the case of van der waals gases , we obtain u_i(u , ) = ( u + _ j ( n_j,0+_j ) ^2 ) - ( n_i,0+_i ) ^2 . [ redvdw ]let us now investigate the geodesic equations . to construct the metric of the equilibrium manifold , we can use the thermodynamic potential or any other potential that can be obtained from by means of a legendre transformation ( basically , ) .the results do not depend on the choice of , because the geometric properties of in gtd are legendre invariant .if we take the potential , the geodesic equations can be written in general as   according to the picard - lindelf theorem , given an initial value , i.e. , , , and , if is smooth -as in the preceding cases- , the solution to this equation exists and is unique . in the case of a chemical reaction , the value of is fixed by the initial equilibrium state of the reaction , but the value of remains free . in the case of ideal gases , in which is proportional to , the initial values of the second variable and are fixed by the conditions of the chemical reaction . in the case of more general species ,a legendre transformation must be performed such that becomes the second independent variable and the initial values are and .we have seen in the examples above that the final equilibrium state of the reaction does not depend on and that the numerical integrator detects a singularity " at that point .it turns out that this corresponds to a coordinate singularity of the metric . to see this, we calculate the component of the general metric , and obtain [ componente ] g _ = ( ) ^-1 = ( _ i ) ^-1 ( _ i ) .this expression is valid for and all the thermodynamic potentials that can be obtained from by means of any legendre transformations -except those which change the role of - .it is easily seen that as soon as the chemical - reaction equilibrium condition is satisfied , the denominator goes to zero and , thus , the metric is not well defined .this means that at the final equilibrium point the coordinates are not appropriate to describe the equilibrium manifold .notice that this result is completely general since it does not depend on the particular reaction that determines the potential .the fact that the final equilibrium state is characterized by a coordinate singularity allows us to perform an analytical investigation of that particular point .consider , for instance , the metric ( [ gig ] ) for two ideal gases in the .the component presents a physical divergence when .the solution of this equation _f= with gives which is exactly the value obtained in the numerical investigating of the geodesic equations for this metric .  finally , let us mention an additional invariance property of the gtd approach . to reduce the number of independent variables of the fundamental equation in the case of van der waals gases , we used the relationship ( [ redvdw ] ) for with .however , due to the invariance of the metric under changes of coordinates , we can also use the same relationship as .then , the resulting fundamental equation becomes , once is fixed in accordance with the conditions of the reaction .the corresponding metric can be computed and the geodesic equations can be integrated numerically for and with the same initial conditions for , and and .the resulting geodesics are exactly the same as the ones we obtained with .in this work , we used the formalism of gtd to present a geometric representation of chemical reactions in closed systems . in gtd , all the information about a thermodynamic system is encoded in its equilibrium manifold determined by a metric which is invariant under legendre transformations , i.e , its properties do not depend on the choice of thermodynamic potential .first , we consider the case of a chemical reaction with only two species corresponding either to ideal gases or to van der waals gases . in the case of ideal gases, we found that the equilibrium manifold is flat , independently of the thermodynamic potential . in gtd, a vanishing curvature means that there is no thermodynamic interaction .this agrees with the interpretation from the point of view of statistical mechanics and thermodynamics : with the statistical point of view , because the molecules of each ideal gas behave as free particles " since the hamiltonian contains exclusively the kinetic part ; and with the thermodynamic point of view , because there are no coupling terms in the fundamental equation . in the case of van der waals gases ,the curvature is different from zero , indicating the presence of thermodynamic interaction .this is also in accordance with the statistical approach to the van der waals system , because the hamiltonian contains a potential term which is responsible for the interaction between the molecules of the system .the thermodynamic analysis of the chemical reactions in both cases shows that the final equilibrium state is reached at a particular value of the extent of reaction parameter .the numerical analysis of the geodesics in the corresponding equilibrium manifolds provides exactly the same value of the extent of reaction for the final equilibrium state .it is evident that all the geodesics follow a path pattern that ends at the same equilibrium state , independently of the initial values of and .moreover , the final equilibrium state is always denoted as a singularity by the numerical integrator .  in order to understand our results in a more general fashion, we analyzed a general reaction in the context of gtd .we showed that using the conditions of thermodynamic equilibrium and the laboratory conditions of the reaction , it is always possible to reduce to two the number of dimensions of the equilibrium manifold .this is an interesting result that allows us to describe any chemical reaction as a geodesic curve on a space .finally , it was shown that the metric of the equilibrium manifold possesses a coordinate singularity exactly at that point where the condition for the reaction equilibrium is satisfied .the examples of chemical reactions presented in this work involve only gases .nevertheless , the generalization to include reactions involving solids or liquids is straightforward .indeed , once the fundamental equations of the species are given , we can construct the corresponding massieu - planck potential of the reaction which allows us to consider and as constants .the main conclusions of this work is that to any chemical reaction in a closed system we can associate a equilibrium manifold , and that any chemical reaction can be represented as a geodesic in which the initial state is determined by the initial conditions of the reaction and the final state corresponds to a coordinate singularity of the thermodynamic metric .we would like to thank the members of the gtd - group at the unam for fruitful comments and discussions .this work was supported by conacyt - mexico , grant no .166391 , and by cnpq - brazil .one of us ( dt ) would like to thank conacyt , cvu no .442828 , for financial support .h. quevedo and a. ramirez , _ a geometric approach to the thermodynamics of the van der waals system _ in proceedings of mario novello s 70th anniversary symposium , edited by n. pinto - neto and s. e. perez - bergliaffa ( 2012 ) pp .301 - 313 ; arxiv:1205.3544

Abstract: we use the formalism of geometrothermodynamics to describe chemical reactions in the context of equilibrium thermodynamics . any chemical reaction in a closed system is shown to be described by a geodesic in a manifold that can be interpreted as the equilibrium space of the reaction . we first show this in the particular cases of a reaction with only two species corresponding to either two ideal gases or two van der waals gases . we then consider the case of a reaction with an arbitrary number of species . the initial equilibrium state of the geodesic is determined by the initial conditions of the reaction . the final equilibrium state , which follows from a thermodynamic analysis of the reaction , is shown to correspond to a coordinate singularity of the thermodynamic metric which describes the equilibrium manifold .


the renowned italian physicist enrico fermi was famous for his ability to make reliable estimates , even with data that would have seemed insufficient to many people .one of the best known examples of this is the estimate he made of the power of the first atomic bomb detonated on july 16 , 1945 at the new mexico desert , measuring the distance travelled by a few scraps of paper that he dropped to the ground while participating as an observer of the explosion .later , when he served as professor at the university of chicago , fermi used to pose such problems to his students as a teaching method .the most famous was to estimate the number of piano tuners in chicago at that time . making a series of reasonable assumptions , such as the number of people living in chicago , the number of people living on average in every household , every how many houses there is a piano , how often a piano should be tuned , how long it takes for a tuner to do his work , etc ., he could effectively estimate the number of piano tuners in chicago , obtaining a result that compared reasonably well with those contained in the telephone directory .  in the first year of the physics teacher program at the unrnwe frequently face the students with everyday problems , which can sometimes be solved  la fermi " .we find it is a very good method to help students sharpen their imagination as well as their observation skills , and at the same time is a training in logical reasoning .within this type of problem we recently raised the possibility of estimating the power developed by the puyehue - cordn caulle volcanic system , near the argentine - chilean border , in its eruption of june 4 , 2011 at 17:00 utc ( 14:00 local time , gmt-3 ) . to do thiswe consider only the first eruption , which covered with ashes and sand the some small villages in chile and the cities of san carlos de bariloche ( , ) , villa la angostura ( , ) and surrounding areas in argentina . in figure[fiaqua ] the first stage of the eruptionis shown , as it has been recorded from space by nasa aqua satellite .  ]the working hypotheses that we use are :  1 .  area covered by sand and dust . + according to photographs taken by nasa terra satellite on june 5 , 2011 , wind conditions during the eruption day made the cloud of sand and ash follow almost a straight line running along the nahuel huapi lake , in the wnw - sse direction .although the fallout was only partially visible on water , it is clear that it covered the entire area of the lake .therefore we use this area as a parameter to estimate the amount of ash fall . as a working hypothesiswe assume the area covered by ash and sand to be about three times the area of nahuel huapi lake , .+  + in figure[terra ] the nasa terra satellite image for 05 jun 2011 at 13:45 utc is shown .the blue profile corresponds to the coastline of the lake , and in red the area covered by sand and ashes is highlighted . as a guide to the eye , a rectangle of area is also indicated . to determine ,the following data was used : the surface of the lake is ; adding the surfaces of the largest islands , victoria and huemul , totalling , we find . so the area covered by ash would be .the picture has been post processed by using the google earth software .thickness of the sand layer .+ in line with the above hypothesis on the covered area , we assign a value to the average thickness of the layer of sand and ash. it is known that a layer over m covered villa la angostura while east of bariloche the layer was less than m thick . on the other hand reports indicate an accumulation of more than m in the town of ingeniero jacobacci ( , ) 210 km eastward from bariloche .this corresponds to ash and sand that did not fell into the area but which must be included in the computation since its elevation affects the energy balance of the volcano . to account for the total energy cost we assume an average height of m . covering the chosen area .duration of the first eruption .+ the ash fall in bariloche started at 16:30 local time ( art , gmt-3 , 19:30 utc ) and ended five hours later , at 21:30 art , so the total time of the eruption first phase is 5 hours , .4 .  height reached by the cloud .+ the height of interest is the difference where m is the height of the volcano .we have m , an average value as different components of the column reached varying heights , the lightest getting up to m .density of the mixture of sand and dust .+ the density of the sand and ash fell in bariloche was determined using household items : a measuring cup used to measure the flour or sugar and a kitchen scale .the result was .this value , lower than the density of water , is explained because ash and pumice , which floats on water , are mixed with denser components that tend to sink into water .a simple volumetric calculation shows that the volume of sand and ash fall is considering that a cargo truck can carry about , the amount of sand fall is equivalent to i.e. , more than twenty four million trucks of sand .  an alternative way to visualise the amount of sand consists of calculating the height it would reach if it were placed in a square area of side by stacking a pyramid of height . the volume of the pyramid must match the estimated volume of sandhowever , as the angle of inclination of the pyramid should not exceed the `` angle of repose '' of the sand , approximately , there is a link between the height and the base .we have the relations {\frac{6v}{\tan \theta } } \\  h&=&\frac{l}{2}\tan \theta\end{aligned}\ ] ] with .it follows this means that the fallen sand could be stacked in a square field km wide ( with an area of ) , making a m high pyramid . as a visual reference we note that the area of the emergent of huemul island , in bariloche is about ha ( ) , and its height certainly less than the m we obtained here .   to determine the mass of sand fall , we multiply the calculated volume by the measured density , to obtain i.e. about 100 million of metric tonnes .   the energy required to raise this mass of sand to mcan be estimated from the potential energy acquired when it reaches its maximum height .we have where is the acceleration of gravity .thus we have that is more than one thousand kilotons of tnt equivalent . for comparison , we mention that the 2011 earthquake in japan had a magnitude of in the richter scale and released an energy of ( of tnt equivalent ) .another reference point is given by the total energy produced in a country .for example , the installed electric power in argentina is and so , the energy produced during a day is .thus the energy released by the first phase of the eruption is equivalent to that produced in the whole country in .the magma rises through the conduit of the volcano mixed with fumes of molten rock and water vapour , which results in a turbulent two phase flow .the velocity of the magma exiting the crater can be coarsely estimated from the energy conservation law , since the flow , at least for the denser components , is ballistic .the ashes reaching the top of the column probably ascend the last section by convection . for that reason we took an average altitude of m , a moderate value compared to the m or even m that have been reported . neglecting air friction, the kinetic energy of fluid at the crater must be equal to the potential energy reached in the final ascent , from this it follows that that is , nearly the speed of sound in air .this high speed of the outgoing material determines its rapid cooling , which explains the amorphous structure and the presence of micro crystals in the ejecta since production of larger crystals requires slow crystallisation .   assuming that the first eruption lasted , i.e. , it is possible to estimate the power generated by the volcano , which turns out to be by comparing this result with the installed electric power in argentina mentioned above, the ratio obtained is the power developed by the volcano is times the installed power in argentina .we can also compare our result with the worldwide installed electrical power , which is .a percentage comparison results in so the power developed by the puyehue volcano is equivalent to about 2% of the global electric power .in the problem of piano tuners , fermi could compare the estimated result with the answer obtained from a query in the telephone directory .the discrepancy could be attributed to the fact that some tuners had no phone or were not listed as such in the directory , or to an error in the various estimates .but this last point is part of the game : the idea is not to obtain an accurate result , but to reach a reasonable approximation . as suchwe understand a discrepancy not exceeding a certain limit , say half an order of magnitude , or a factor of five .  in what follows we comment on each of the scenarios we used :  1 .area covered .+ in the eruption of june 4 , 2011 ash and sand covered an area larger than the one assumed in this work .that could mean that we underestimate the energy and power of the volcano .we tried to compensate for material falling beyond the chosen zone , by assuming a uniform thickness of cm in the reference area .average thickness of the layer of sand and ash .+ it is known that in the vicinity of the volcano on the chilean side and in villa la angostura the layer of sand and ash reached cm , while further east it did not exceeded cm .we believe that the figure of cm distributed uniformly over an area equal to three times the area of the nahuel huapi lake ( over ) is appropriate as an average estimate .3 .  estimated time of activity . +the fall of sand and ashes in bariloche ceased at 00:30 utc ( 21:30 art ) , that is the first eruption lasted five hours .the thunder caused by electric shock lasted a total of approximately indicating that volcanic activity continued until then .this does not affect the estimation of the energy developed in the first five hours .4 .  height reached by the cloud .+ reports on june 4 indicated a height of the column of up m above sea level .as the column included different density components , we believe our assumption m for the thicker components of the plume is a reasonable estimate for our purposes .density of the mixture of sand and ashes .+ the density measurement was made with a sample collected within hours of the onset of the rash .samples taken a few days later gave higher density values , probably because rain water dissolved or dragged away part of the components and swept them away .6 .  output speed .+ we are not aware of measurements of the output speed of the magma in the volcano being made , but specialised authors have studied the supersonic flow of magma in volcanoes in some typical situations of the so - called explosive pyroclastic eruptions ( see for example ) , as the puyehue - cordn caulle eruption of june 4 , 2011 .an eruption of this type occur in the year 79 of the christian era at the vesuvian volcano , inducing the tragic disappearance of the city of pompeii .this eruption had two distinct phases : first a _phase , where material was ejected in a tall column , spread in the atmosphere and fell to earth like rain ; followed by a _ pelan _ phase where material flowed down the sides of the volcano as fast - moving avalanches of gas and dust , called pyroclastic flow ( pyroclasts are rock fragments formed by a volcanic explosion or ejected from a volcanic vent ) .the solution of the present problem indicates that using knowledge normally available to high school students , and introducing a series of reasonable assumptions , it is possible to solve approximately a problem resulting from a natural event that affects everyday life .andrew w. woods and sally m. bower , _ the decompression of volcanic jets in a crater during explosive volcanic eruptions _ ,earth and planetary science letters , volume 131 , issues 3 - 4 , april 1995 , pages 189205 .mitchell , volcanic eruption mechanisms , _ coupled conduit flow and shape in explosive volcanic eruptions , _ journal of volcanology and geothermal research , volume 143 , issues 13 , may 2005 , pages 187203  adriana bermdez y daniel delpino , _ la actividad del complejo volcnico puyehue - cordn caulley su impacto sobre el territorio de la repblica argentina _ , http://www.conicet.gov.ar/new_noticias/noticias.php?id_noticia=7192&tipo=6&nota_completa=yes[primer informe ] , conicet , june 15 , 2011 .adriana bermdez y daniel delpino , _ la actividad del complejo volcnico puyehue - cordn caulle y su impacto sobre el territorio de la repblica argentina _ , http://www.conicet.gov.ar/new_noticias/noticias.php?id_noticia=7295&nota_completa=yes&tipo=6[segundo informe ] , conicet , june 28 , 2011 .

Abstract: on june 4 2011 the puyehue - cordn caulle volcanic system produced a pyroclastic subplinian eruption reaching level 3 in the volcanic explosivity index . the first stage of the eruption released sand and ashes that affected small towns and cities in the surrounding areas , including san carlos de bariloche , in argentina , one of the largest cities in the north patagonian andean region . by treating the eruption as a fermi problem , we estimated the volume and mass of sand ejected as well as the energy and power released during the eruptive phase . we then put the results in context by comparing the obtained values with everyday quantities , like the load of a cargo truck or the electric power produced in argentina . these calculations have been done as a pedagogic exercise , and after evaluation of the hypothesis was done in the classroom , the calculations have been performed by the students . these are students of the first physics course at the physics and chemistry teacher programs of the universidad nacional de ro negro .


adaptive filtering algorithms deal with real - time learning scenarios , in which the environment is often nonstationary . in general , these algorithms need to fulfill three basic requirements : 1 ) to sequentially learn from each observation ; 2 ) to be adaptive to changing environments ; and 3 ) to be computationally efficient . among many existing algorithms that fulfill these requirements , one that has stood the test of time is the celebrated least mean squares ( lms ) algorithm .this algorithm has several interesting properties , in particular its inherent computational simplicity , and its implicit tracking ability despite its assumption of stationarity .  inspired by the success of the lms algorithm ,a `` kernelization '' has been recently proposed under the name kernel least mean squares ( klms ) algorithm  .the klms inherits many desirable properties of lms and extends it to a large class of nonlinear filtering algorithms .nevertheless , it has certain limitations that arise from its formulation as an adaptive filter in a possibly infinite dimensional feature space .specifically , if implemented naively , the representation of the filter grows linearly with the number of data samples processed .moreover , both the lms and the klms explicitly minimize the squared error between the desired and the estimated observation values , hence , they can not be naturally applied to problems with discrete observations such as class labels .to summarize , klms , in its current format ,  1 .  can not be extended to tackle discrete observations , 2.  grows indefinitely with new observations , and 3 .does not provide an explicit understanding of its tracking ability ( see below ) .the available adaptive filters in the statistical signal processing literature can be broadly categorized as 1 ) filters derived from a ( stationary ) regression framework , such as lms and recursive least squares ( rls ) , and 2 ) filters derived from a ( nonstationary ) state - space framework such as the kalman filter ( fig .[ fig : graphical : model ] ) .these two classes of algorithms are both successfully applied to nonstationary systems , and occasionally show similar computational characteristics , e.g. , extended rls can be estabilished as a special case of the kalman filter  .we seek a principled explanation based on the bayesian filtering framework of how lms achieves tracking despite its formulation under the assumption of stationarity , since it would allow us to systematically address the limitations of the klms discussed earlier .the connection we seek is related but distinct from that of bayesian approaches recently explored by  for kernel rls ( krls ) ; specifically , it was shown that a recursive filtering implementation of the bayesian regression framework ( gaussian processes ) naturally leads to a formulation that is equivalent to kernelization of rls .they have explicitly introduced two types of forgetting methods to enable tracking from which we were inspired for extending the klms in this paper but the forgetting is not incorportated within the bayesian framework .is assumed to have the same relation as in the classic regression setting .original derivation of lms is given in this context .( b ) non - stationary model .the weight evolves over time hence the relation between diffusion with parameter .the kalman filter is derived under this model .we show the klms can also be derived from the nonstationary model ., width=307 ]  in this paper , we show that lms ( and klms ) can indeed be derived as an approximation of a state - space based bayesian filtering ( section [ sec : bayes ] ) . in order to achieve a low computational complexity , though, only the mode of the posterior distribution can be estimated and retained for each sample .this new interpretation allows us to derive extensions of the klms algorithm by tweaking the underlying state - space and observation models ( section [ sec : forgetting ] and [ sec : observation ] ) . here, we extend the klms algorithm to integer and binary - valued observations , and also introduce a forgetting factor in order to improve its tracking ability .we furthermore illustrate each extension with a simple example to demonstrate its performance in tracking nonstationary signals , and compare with existing methods such as the quantized klms ( qklms )  and naive online regularized risk minimization algorithm ( norma )  . some interesting properties of these extensions are that each algorithm learns by assigning a coefficient for each new observation , and that the time complexity of the -th iteration has linear cost , , in both space and time , as is the case for the klms .both widrow & hoff s lms , and klms are derived from mean squared error cost function  which is prevalent in traditional signal processing .the filtering setting assumes a linear model in the _ feature space _ where is the feature vector associated with the input vector , and is the vector representation of the filter in a hilbert space .the following derivation holds for both lms and klms , taking into account that for lms the feature space is the ( euclidean ) input space itself , i.e. , , while klms uses a ( potentially ) infinite dimensional ( reproducing kernel ) hilbert space induced by a positive definite kernel where  .the mean squared error is defined as , ,\end{aligned}\ ] ] where and are the random vector and variable for the input signal and the desired output , respectively .note that this cost function corresponds to a regression problem corrupted by gaussian noise assuming independence , illustrated in fig .[ fig : graphical : model]a , where given a set of input - output data pairs , the negative log - likelihood is given by where is the noise variance .in the limit of large number of _ iid _ samples , due to the law of large numbers , it converges to its expectation which is the mean squared error ( up to a factor and a constant ) .therefore , the ( asymptotic ) maximum likelihood solution under this model coincides with the minimum mse ( mmse ) solution .the basic steepest descent learning rule has the form .\end{aligned}\ ] ] since the cost function is convex , it will converge to the mmse solution for a sufficiently small learning rate . to make an online learning rule ,a stochastic gradient descent is used in practice . in particular , the learning rule of the lms algorithm is obtained by dropping the expectation from , which yields hence , after processing samples , the prediction for the next sample is given by , where is the error for each sample . for klms ,the prediction can be directly computed from the samples despite , since .the stochastic gradient descent algorithm is guaranteed to convergence ( almost surely ) to the global optimal solution under stationary and ergodic observations if a proper step size scheduling is used ( e.g. , and ) .however , the tracking capability of lms / klms is dependent on the step size ; if the step - size were annealed , it would be tracking less efficiently as more samples are seen .therefore , to have constant tracking , step size is not annealed in practice , that is , .then , for the price of non - zero misadjustment , the algorithms can surprisingly learn continuously from new samples , and overwrite what was learned before .note that this `` hack '' disconnects the algorithm from the graphical model fig .[ fig : graphical : model]a which inherently assumes a stationary data generation process . in the following section ,we show how this tracking ability can be derived from first principles .a slowly changing system can be explicitly described by a probabilistic model with latent dynamics on the parameter . in such a model ,each parameter associated with each sample or time is considered as an interdependent random ( latent ) variable , as illustrated in fig .[ fig : graphical : model]b .our goal is to show that the klms is an approximate sequential inference for .we start with a diffusion process as a reasonable model for nonstationarity : where denotes a gaussian distribution in the feature space , and is the variance of diffusion on each direction . the likelihood model is assumed to be a linear gaussian model , similar to the stationary case , where is the observation noise variance .we remark that the conditional distributions and for the finite dimensional feature space is a special case of the kalman filter model with linear dynamics .   if we wish to ( recursively )infer the posterior weight distribution , we only need the mean and covariance , since we assume gaussianity in this model  . assuming , a single linear gaussian observation results in a one step evolution of the posterior as another gaussian , with .\end{aligned}\ ] ] this recursion can be solved efficiently , and the solution is known as the extended recursive least squares algorithm . however , it requires a quadratic number of operations in terms of the dimension of the feature vector for updating the ( inverse ) covariance matrix . in case of an infinite - dimensional feature space , the feature vector dimension grows linearly with the number of observations , rendering this approach prohibitive .therefore , in order to obtain a linear time complexity algorithm , we assume the posterior to be concentrated around the maximum .in other words , we approximate the posterior as a delta function at the maximum a posteriori ( map ) estimate before inferring .below , we show the steps for online inference rules using this approximation .first , the approximation is equivalent to assuming an isotropic gaussian around the map estimate for the previous sample . bayes rule , the posterior weight distribution is , where the parameters for the posterior are , .\end{aligned}\ ] ] this can be simplified using the matrix inversion lemma , where the learning rate is determined by the diffusion - to - noise ratio .this is very similar to the normalized lms ( nlms ) update rule , although not identical . if the kernel is normalized , such that , then it can be simply rewritten as , where .note that the stochastic gradient derivation is identical to the approximate bayesian learning rule ; we have rederived klms with a state - space model .also , note that , thus we have a frequentist convergence guarantee of the weight vector to the optimal weight vector in mean in a stationary environment , i.e. , \rightarrow { \mathbf{w}}^\ast ] and = \sigma^2_q ] , and klms is not constrained at all .pruning is accomplished in fklms and norma by dropping the oldest coefficient .a squared exponential kernel with kernel size of is used for all the algorithms .the mse is computed for a one - step prediction , and averaged out over simulations .we scan several values of the parameters , and report the parameter with least average normalized mse over the iterations .asymptotic normalized mse is estimated from 800 samples after time step 200 ( error bars indicate 2 standard error ) .interestingly , we observe that klms and qklms , which lack a forgetting mechanism , obtain weaker tracking performance . ]note that the learning rule  can be expanded , and rewritten as , where is a scalar corresponding to the coefficient at the learning step .we can see that each effective coefficient for each shrinks geometrically over time .thus , the effect of older observation to the current weight estimate is small in general .note , however , that the algorithm forgets not by making the covariance larger as in the kalman filter , but by changing the the mean .  like norma , the forgetting dynamics klms can be interpreted as introducing a regularization for the weight vector .however , as shown in , klms is self - regularizing and hence such extra regularization is usually not necessary .however , it provides a significant practical benefit for maintaining a compact representation we can prune the component that are too small to have any effect .a simple strategy of pruning is to drop effective coefficients below a small pre - specified threshold .then , assuming a stationary distribution over the new coefficients , the forgetting factor and the threshold control the expected representation length .one can also maintain a fixed budget by removing the oldest coefficient after a predefined number of centers is reached .we employ the latter strategy in the experiment shown in fig .[ fig : tracking : fklms ] .gaussian observation model  is widely used for continuous observations , however , it is inappropriate where the observations are natural numbers , or binary labels . in this section ,we extend klms by replacing the gaussian observation model with other distributions in the exponential family .poisson likelihood is widely used when the observations are natural numbers : .for example , in neuroscience , neural response is often quantified by the number of spikes , and tracking how the neural code changes during experiment is of great importance  .we use the canonical inverse link function ( exponential ) for the poisson distribution to map the linear ( or nonlinear ) function from the input to the non - negative rate parameter , i.e. , to derive the adaptive filtering algorithm , once again , we approximate the current state given the previous observations as , for which the log prior is , therefore , using bayes rule , the posterior at time is , where irrelevant constants are omitted .we need to maximize this log - posterior over to estimate .the map estimate must satisfy , the stationary condition , which implies , we observe that the solution of can be expressed as , where is a scalar , and therefore , we can rewrite the log - posterior as , where , and we have assumed a normalized kernel for simplicity .thus , we reduce the problem of finding an infinite dimensional weight vector to a one dimensional optimization . although , there is no analytical solution , the cost function is strictly concave and therefore , its maximum can be easily found by existing optimization tools .the complexity of this algorithm is still , with a constant overhead for solving a concave maximization problem at each step .we demonstrate its performance on a neurally inspired example in fig .[ fig : tracking : poisson ] .a typical nonlinear response function ( tuning curve ) is set to shift its center slowly , creating a non - stationary tracking problem .the poisson - klms extension correctly tracks , and maintains a small mse throughout the experiment .   where constantly drifted degrees during the iterations .we measure the normalized estimation error between the true tuning curve and the estimated curve .insets show the actual function estimate at 25 , 50 , 100 , 500 time steps .gray lines show 11 repeats of the experiment , and the dark curves correspond to their average .the kernel was and . ]   similarly , bernoulli likelihood is widely used when the observations are binary labels , such as in a classification problem .here we only address the binary classification problem , while the generalization to multi - class classification is certainly feasible , and straightforward .we use the inverse canonical link function ( logistic ) for the bernoulli distribution to map the linear ( or nonlinear ) function from the input to the non - negative probability score between $ ] , i.e. , where , and is the bernoulli distribution with probability of success .then the posterior log - liklihood can be written as , once again , we need to maximize this likelihood to get . taking derivative of this function , we get , as in the poisson case , it can be observed that the stationary point for which the gradient is zero , can be expressed as .so , we need to only solve for by maximizing where .the cost function is again strictly concave , and thus , the optimization problem can be solved efficiently . in fig .[ fig : tracking : logistic ] , we demonstrate the effectiveness of this approach in tracking a shifting binary classification boundary .   to ) .three contours represent 0.25 , 0.5 , and 0.75 probability of the posterior .the kernel was and . ]it should be noted that both poisson - klms and bernoulli - klms do not have an explicit learning rate parameter .this is because the observation model is not gaussian , where the learning rate parameter is simply the ratio between the diffusion variance and the noise variance .however , in these extensions the diffusion variance plays a similar role ; the lower the , the slower the adaption process .we acquired data from a wireless communication test bed that is used to evaluate the performance of digital communication systems in realistic indoor environments .the platform is composed of several transmit and receive nodes , each one including a radio - frequency front - end and baseband hardware for signal generation and acquisition .the front - end also incorporates a programmable variable attenuator to control the transmit power value and therefore the signal saturation .a more detailed description of the test bed can be found in . using the hardware platform, we transmitted clipped orthogonal frequency - division multiplexing ( ofdm ) signals centered at ghz over real frequency - selective and time - varying channels , with normalized doppler frequency around .the transmit amplifier was operated close to saturation . in this experimentthe transmitted and received signals are used to track the variations of the nonlinear channel .we compare 4 algorithms with hyperparameters tuned using the first 500 samples .[ fig : tracking : testbed ] displays the one - step ahead prediction normalized mean squared error ( nmse ) of the tracking experiment .average nmse are : nlms , norma , qklms , fklms  db . fklms and norma used 500 total basis functions .qklms and fklms show similar performances better than norma and ( linear ) nlms .in this paper , we derived a family of linear time and space complexity kaf algorithms from bayesian filtering by maintaining only the map solution at each iteration and discarding the posterior distribution ( summarized by the covariance ) .one of the basic resulting algorithms is identical to klms , which is simple and practical .the tracking ability of lms / klms is usually understood by its stochastic nature that allows it to continually adjust itself to the non - stationary environment .we provide an alternate explanation of this mechanism by showing the klms can also be seen as an approximation to state - space modeling which possesses explicit tracking abilities .our framework allows flexibility in the state - space models which can be used to induce forgetting behavior , as well as novel observation noise models , such as poisson and bernoulli .  the optimal nonlinear bayesian filtering for gaussian diffusion dynamics , given by either eq .  or eq . , and a gaussian observation model can be iteratively solved by extended kernel recursive least squares algorithm  . for a general dynamics and a general likelihood such as , or , we often do not have a closed form iterative solution , and one must resort to slower sampling based inference such as sequential monte carlo algorithm or approximate inferences .exact optimal solutions are impractical due to its high computational cost .hence , certain approximations must be made ; for example , uses a low - dimensional subspace approximation .we have derived kafs for natural number and binary observation , and it can be also extended to multi - class by using multinomial - logistic model .however , we do not have convergence results for poisson and logistic variants as in the gaussian version where it can be shown that for the solution converges in mean .our algorithms have a few hyperparameters that need to be set ahead ; the kernel parameters , the diffusion variance , and the likelihood parameters .we have not presented a formal way for choosing those hyperparameters , which plays a crucial role in the speed of convergence and the tracking ability of the algorithm . if training time series is available , one can use expectation - maximization or sampling methods to find appropriate parameters  .we leave these as future work .j. gutirrez , .gonzlez , j. prez , d. ramrez , l. vielva , j. ibez , and i. santamara .frequency - domain methodology for measuring mimo channels using a generic test bed . ,60(3):827838 , march 2011 .

Abstract: the kernel least mean squares ( klms ) algorithm is a computationally efficient nonlinear adaptive filtering method that `` kernelizes '' the celebrated ( linear ) least mean squares algorithm . we demonstrate that the least mean squares algorithm is closely related to the kalman filtering , and thus , the klms can be interpreted as an approximate bayesian filtering method . this allows us to systematically develop extensions of the klms by modifying the underlying state - space and observation models . the resulting extensions introduce many desirable properties such as `` forgetting '' , and the ability to learn from discrete data , while retaining the computational simplicity and time complexity of the original algorithm .