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_{i}_{i}_{j}

The first non-classical entropy was proposed by Rényi in 1960 [

where _{i}^{*}_{h}^{*}_{h}_{i}

These divergences have the form of the relative entropy or, in the thermodynamic terminology, the (negative) free entropy, the Massieu-Planck functions [^{*}.

After 1961, many new entropies and divergences were invented and applied to real problems, including Burg entropy [^{*}

The idea of Bregman divergences [

if _{F}_{F}_{F}_{F}

The divergence between the current distribution and equilibrium should monotonically decrease in Markov processes. It is the ultimate requirement for use of the divergence in information processing and in non-equilibrium thermodynamics and kinetics. In physics, the first result of this type was Boltzmann’s

In his well-known paper [_{h}^{*}) monotonically decreases in Markov processes (he gave the detailed proof for the classical relative entropy and then mentioned that for the _{h}_{ij}

where

is the ergodicity contraction coefficient,

Under some additional conditions, the property to decrease in Markov processes characterizes the

The dynamics of distributions in the continuous time Markov processes is described by the master equation. Thus, the _{h}^{*}^{*}

For each new divergence we have to analyze its behavior in Markov processes and to prove or refute the

It is obvious that the equilibrium ^{*}_{i}_{j}

These solutions are minima of

Using this general

We consider continuous time Markov chains with _{1}, …, _{n}_{i}_{1}, …, _{n}^{T}

where _{ij}_{ij}_{j}_{i}_{ij}^{n}^{(}^{n}^{−1)} with coordinates _{ij}

It is useful to mention that the ^{*}

is a particular case of master equation for normalized variables

The _{i}_{j}_{i}_{ij} >_{i}_{j}

A digraph is _{i} to A_{j}, then there exists a directed path from A_{j} to A_{i}_{n}

The Markov chain in _{1} distance
_{i}_{j}_{k}_{i}_{j}

One of the paths can be degenerated: it may be

Now, let us restrict our consideration to the set of the Markov chains with the given positive equilibrium distribution
^{*}

where
_{ij}

For the next transformation of master equation we join the mutually reverse transitions in pairs _{i}_{j}^{ji}

Let us rewrite the master

where
_{j}_{i}_{j}_{i}

The reversible systems with detailed balance form an important class of first order kinetics. The

Here,
_{i}_{j}

For the systems with detailed balance the quasichemical form of the master equation is especially simple:

It is important that any set of nonnegative equilibrium fluxes
^{*}

The ^{*}

In other words, for every general system of the form (13) with positive equilibrium ^{*}

Let ^{*}

The inequality d

A convex function on a straight line is a Lyapunov function for a one-dimensional system with single equilibrium if and only if the equilibrium is a minimizer of this function. This elementary fact together with the previous observation gives us the criterion for universal Lyapunov functions for systems with detailed balance. Let us introduce the

_{n} of probability distributions satisfies the partial equilibria criterion with a positive equilibrium P^{*} if the proportion

_{n} of probability distributions is a Lyapunov function for all master equations with the given equilibrium P^{*} that obey the principle of detailed balance if and only if it satisfies the partial equilibria criterion with the equilibrium P^{*}.

Combination of this Proposition with the decomposition theorem [

_{n} of probability distributions is a Lyapunov function for all master equations with the given equilibrium P^{*} if and only if it satisfies the partial equilibria criterion with the equilibrium P^{*}.

These two propositions together form the general

^{*} is a necessary condition for a convex function to be the universal Lyapunov function for all master equations with detailed balance and equilibrium P^{*} and a sufficient condition for this function to be the universal Lyapunov function for all master equations with equilibrium P^{*}.

Let us stress that here the partial equilibria criterion provides a necessary condition for systems with detailed balance (and, therefore, for the general systems without detailed balance assumption) and a sufficient condition for the general systems (and, therefore, for the systems with detailed balance too).

The simplest Bregman divergence is the squared Euclidean distance between ^{*}

For the Itakura-Saito distance (

If the single equilibrium in 1D system is not a minimizer of a convex function _{i}_{j}_{i}_{j}

If _{i}_{j}

The partial equilibria criterion allows a simple geometric interpretation. Let us consider a sublevel set of _{n}_{h}_{n} | H_{h}_{n}_{i}_{j}_{ij}_{h}_{ij}_{ij}_{h}_{h}

We illustrate this condition on the plane for three states in _{1}_{;}_{2}_{;}_{3} and _{1}_{;}_{2}_{;}_{3}. For each point _{i}_{i}_{jk}

The points of intersection _{1}_{;}_{2}_{;}_{3} and _{1}_{;}_{2}_{;}_{3} cannot be selected arbitrarily on the lines of partial equilibria. First of all, they should be the vertices of a convex hexagon with the equilibrium ^{*}_{ij}_{ij}_{H}_{(}_{P}_{)}. If we apply this statement to _{i}_{i}_{ij}_{ij}_{1}_{;}_{2}_{;}_{3} and _{1}_{;}_{2}_{;}_{3}. They produce a six-ray star that should be inscribed into the level set.

In

All the ^{*}

Conic combination: if _{j}^{*}_{j}

Convex monotonic transformation of scale: if ^{*}^{*}

Using these operations we can construct new universal Lyapunov functions from a given set. For example,

is a universal Lyapunov function that does not have the

The following function satisfies the partial equilibria criterion for every

It is convex for 0 _{n}

Several formalisms are developed in chemical kinetics and non-equilibrium thermodynamics for the construction of general kinetic equations with a given “thermodynamic Lyapunov functional”. The motivation of this approach “from thermodynamics to kinetics” is simple [

GMAL is a method for the construction of dissipative kinetic equations for a given thermodynamic potential

The _{1}, …, _{n}

A

where _{ρi}, β_{ρi}_{ρ}, β_{ρ}_{ρi}, β_{ρi}

A _{ρ}_{ρ}_{ρ} – α_{ρ}

that is, “gain minus loss” in the _{ρ}_{ρ}_{ρ}

One of the standard assumptions is existence of a strictly positive stoichiometric conservation law, a vector _{i}_{i} >

A nonnegative extensive variable _{i}_{i}_{i}_{i}_{i}_{i}_{i}

Let us consider a domain _{1}, …, _{n}_{i}_{B}

The dual variables, potentials, are defined as the partial derivatives of

This definition differs from the chemical potentials [

For each reaction, a _{ρ}

where
_{ρ}

In the standard formalism of chemical kinetics the reaction rates are intensive variables and in kinetic equations for

A nonnegative extensive variable _{i}_{i}_{i}_{i}_{i}_{i}

The kinetic equations for a homogeneous system in the absence of external fluxes are

If the volume is not constant then the equations for concentrations include

The classical Mass Action Law gives us an important particular case of GMAL given by

where _{i}_{i}

For the perfect entropy function presented in

and for the GMAL reaction rate function given by (21) we get

The standard assumption for the Mass Action Law in physics and chemistry is that ^{*}_{ρ}= φ_{ρ}

Thus, the following entities are given: the set of components _{i}_{ρ}

An auxiliary function

With this function,

The auxiliary function

which is the initial mechanism when λ=1 the reverted mechanism with interchange of

It is easy to check that

The inequality

is _{ρ}_{ρ}

The detailed balance condition consists of two assumptions: (i) for each elementary reaction

After this joining, the total number of stoichiometric equations decreases. We distinguish the reaction rates and kinetic factors for direct and inverse reactions by the upper plus or minus:

The kinetic equations take the form

The condition of detailed balance in GMAL is simple and elegant:

For the systems with detailed balance we can take

M. Feinberg called this kinetic law the “Marselin-De Donder” kinetics [

Under the detailed balance conditions, the auxiliary function

The explicit formula for

A convenient equivalent form of

where

is a normalized

The detailed balance condition reflects “microreversibility”, that is, time-reversibility of the dynamic microscopic description and was first introduced by Boltzmann in 1872 as a consequence of the reversibility of collisions in Newtonian mechanics.

The complex balance condition was invented by Boltzmann in 1887 for the Boltzmann equation [

Formally, the complex balance condition means that

Let us consider the family of vectors {_{ρ},β_{ρ}_{1}, …, y_{q}

We can rewrite

The Boltzmann factors

This is the general

It is easy to check that for the first order kinetics given by

The complex balance conditions defined by _{1}, …, _{q}_{ρ}_{ρ}

Let us assign to each edge (_{ρ}, A_{ρ}, A_{ρ}.

Let us use for the vertices the notation Θ_{j}_{j}_{j}, A_{lj}_{j} →_{l}

The _{il}

The graph of the transformation of complexes cannot be arbitrary if the system satisfies the complex balance condition [

First of all, let us formulate Kirchhoff’s first rule (42) for subsets: if the digraph of transformation of complexes satisfies

where _{ΦΘ} is the positive kinetic factor for the reaction Θ _{ΦΘ} =_{j}, A) =

We say that a state Θ_{j}_{k}_{k}_{j}_{k}_{j}_{i}_{↓} be the set of states reachable from Θ_{i}_{i}_{↓} has no output edges.

Assume that the edge Θ_{j}_{i}_{j}_{i}_{↓}. Therefore, the set Ω = Θ_{i}_{↓} has the input edge (Θ_{j} →_{i}

This property (every edge is included in a simple cycle) is equivalent to the so-called “weak reversibility” or to the property that every weakly connected component of the digraph is its strong component.

For every graph with the system of fluxes, which obey Kirchhoffs first rule, the cycle decomposition theorem holds. It can be extracted from many books and papers [

Let us consider a digraph _{i}_{ij} ≥_{ij}

_{il}

Let a function _{+} be an extreme ray of _{ij} >_{ij} =_{j}_{+1}_{j}_{1}, _{k}

Assume that supp

This decomposition theorem explains why the complex balance condition was often called the “cyclic balance condition”.

The class of systems with detailed balance is the proper subset of the class of systems with complex balance. A simple (irreversible) cycle of the length

For Markov chains, the complex balance systems are all the systems that have a positive equilibrium distribution presented by

In nonlinear kinetics, the systems with complex balance provide the natural generalization of the Markov processes. They deserve the term “nonlinear Markov processes”, though it is occupied by a much wider notion [

Nevertheless, in some special sense the classes of systems with detailed balance and with the complex balance are equivalent. Let us consider a thermodynamic state given by the vector of potentials
_{i}_{j}_{i}_{j}_{DB}(
_{CB}_{D}B(
_{CB}

Because of the cycle decomposition (Proposition 4) it is sufficient to prove this theorem for simple normalized cycles. Let us use induction on the cycle length _{2} and the detailed balance condition (35) coincides with the complex balance condition (42). Assume that for the cycles of the length below _{1} → Θ_{2} → _{k}_{1}, Θ_{i}_{i}

At the equilibrium, all systems with detailed balance or with complex balance give Ṅ _{i}_{i}_{1}) is the minimal one and exp(
_{2}) _{1}).

Let us find a kinetic factor _{2} → _{k}_{2}, gives the same Ṅ at the state
_{1} _{k}_{k}_{2} _{1}) exp(
_{1}) _{2} _{k}_{k}_{2} _{1})(exp(
_{1}) _{2})). It is sufficient to equate here the coefficients at every _{i}

By the induction assumption we proved that theorem for the cycles of arbitrary length and, therefore, it is valid for all reaction schemes with complex balance. □

The cone Q_{DB} (

This follows from the form of the reaction rate presented by

Consider GMAL kinetics with the given reaction mechanism presented by stoichiometric _{ρ}_{ρ}_{ρ}

Therefore, nonnegativity is the only a priori restriction on the values of_{ρ}

One Lyapunov function for the GMAL kinetics with the given reaction mechanism and the detailed balance condition obviously exists. This is the thermodynamic Lyapunov function _{i}_{i} =

Assume that we select the thermodynamic Lyapunov function

For a given reaction mechanism we introduce the partial equilibria criterion by analogy to Definition 1. Roughly speaking, a convex function

For each elementary reaction

Definition 2 (Partial equilibria criterion for GMAL).

The partial equilibria criterion is necessary because

For the general reaction systems with complex balance we can use the theorem about local equivalence (Theorem 2). Consider a GMAL reaction system with the mechanism (18) and the complex balance condition.

The theorem follows immediately from Theorem 3 about Lyapunov functions for systems with detailed balance and the theorem about local equivalence between systems with local and complex balance (Theorem 2). □

The general

where

In all versions of the general

For the kinetic Lyapunov functions that satisfy the partial equilibria criterion, we use, actually, a rather weak consequence of convexity in restrictions on the straight lines

A function

In particular, a function

Among many other types of convexity and quasiconvexity (see, for example [_{ρ}_{ρ}

Let _{ρ}|ρ=

Finally, we can relax the convexity conditions even more and postulate _{ρ} | ρ =_{ρ}

Relations between these types of convexity are schematically illustrated in

Many non-classical entropies are invented and applied to various problems in physics and data analysis. In this paper, the general necessary and sufficient criterion for the existence of _{i} ⇌ A_{j}

If an entropy has no

The general

For the reversible Markov chains presented by

These inequalities are closely related to another generalization of convexity, the Schur convexity [

Introduction of many non-classical entropies leads to the “uncertainty of uncertainty” phenomenon: we measure uncertainty by entropy but we have uncertainty in the entropy choice [

It is not possible to compare different entropies without any relation to kinetics. It is useful to specify the class of kinetic equations, for which they are the Lyapunov functionals. For the GMAL equations, we can introduce the dynamic equivalence between divergences (free entropies or conditional entropies). Two functionals _{ρ}

_{ρ}

For all

For the Markov kinetics, the partial equilibria criterion is sufficient for a convex function

The author declares no conflict of interest.

The triangle of distributions for the system with three states _{1}, _{2}, _{3} and the equilibrium
_{i}_{j}_{1} ⇌ _{2} by solid straight lines (with one end at the vertex _{3}), for _{2} ⇌ _{3} and for _{1} ⇌ _{3} by dashed lines. The lines of conditional minima of _{1} ⇌ _{2} (a) for the squared Euclidean distance (a circle here is an example of the ^{*}^{*}

_{2}, _{3}_{i}_{j}_{1}_{;}_{2}_{;}_{3} and _{1}_{;}_{2}_{;}_{3}. (The points _{i}_{i}_{i}^{*}_{i}_{i}_{i}_{j}_{i}_{i}_{i}_{j}_{i}, A_{j}^{ji}_{i}_{i}_{i}_{i}_{j}_{k}^{*}_{i}_{i}_{j}_{k}^{*}_{j}_{k}_{i}_{i}

Monotonicity on both sides of the minimizer

Relations between different types of convexity