Diffusion equations from master equations -- A discrete geometric approach

In this paper, master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology play roles, and master equations are described on graphs that consist of vertexes representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacian is defined as a self-adjoint operator with respect to an introduced inner product. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.


Introduction
Master equations are vital in the study of nonequilibrium statistical mechanics [1], since they are mathematically simple and allow to show relaxation processes towards to equilibrium states [2]. These equations describe the time evolution of the probability that discrete states are found, and they are first order differential or difference equations. In addition, these equations are used in Monte Carlo simulations [3]. Quantum mechanical case can be considered by extending classical systems [4]. Thus there have been a variety of applications in mathematical sciences, and its progress continues to attract attention in the literature [5,6].
Algebraic topology can be applied to a variety of sciences and mathematical engineering. Such topological approaches to master equations exist in the literature [7,8,9]. Not only master equations, but also random walks on lattices [10], electric circuits [11,12], and so on, have been studied from the viewpoint of algebraic topology. By introducing inner products for functions on graphs, one can define adjoint operators and Laplacians as self-adjoint operators [10]. These operators are useful as proven in the literature of functional analysis [13]. It is then of interest to explore how above mentioned operators can be used for master equations. Also, although discrete diffusion equations are derived from master equations in some cases, the condition when such diffusion equations can be derived is not known. Since the knowledge of discrete diffusion equations has been accumulated, clarifying such a condition is expected to be fruitful for the study of master equations.
In this paper master equations are formulated in terms of functions of chains, where states and transition matrices for master equations are described by chains used in algebraic topology. In particular, probability distribution function is regarded as a function of 0-chain, and transition matrix a function of 1-chain, where a discrete state is expressed as a vertex or a 0-chain. After introducing some inner products for functions on chains and current as a function on 1-chain, it is shown that master equations are written in terms of co-derivative of the current, which is equivalent to the following. Claim. Master equations can be written as a form of a continuity equation (See Theorem 3.1 for details). In addition, under the assumption that the detailed balance conditions hold, an equivalence between master equations and discrete diffusion equations is shown, where the Laplacian is constructed by choosing appropriate measures for inner products: Claim. Master equations under the detailed balance conditions are equivalent to discrete diffusion equations (See Theorem 3.2 for details).
By applying this statement, it is shown that probability distribution functions relax to the equilibrium state (Corollary 3.1). By contrast, it is shown that discrete diffusion equations yield master equations (See Proposition 3.1).
These theorems, corollary, and so on should be compared with those in the existing literature. In the study of random walks on lattices, chains, functions and their Laplacians are also used [10]. In the literature the probability distribution functions are identified with functions on 1-chains, which is different to the present formalism. The differences appear since transition matrices are introduced for describing state transitions for the present study. On the other hand, similarities between the present study and existing studies appear due to the use of Laplacians. Laplacian is a self-adjoint operator with respect to an inner product, and brings several properties as well as the case of the standard Riemannian geometry.
In Section 2, some preliminaries are provided in order to keep this paper self-contained. In Section 3, master equations are formulated on graph, and the main claims of this paper and their consequences are provided.

Preliminaries
Let G = (V, E) be a directed graph with V a vertex set and E an edge set. Throughout this paper, every graph is finite (#E < ∞), connected, and allowed to have loop edges. However parallel edges that will be defined later are excluded from this contribution.

Standard operators
In this subsection, most of notions and notations follow Ref. [10]. However, some variants are also introduced.
For a given edge e ∈ E, the inverse of e, the terminus of e, and the origin of e are denoted by e , t(e) ∈ V , and o(e) ∈ V , respectively. In this paper, e ∈ E for any e ∈ E is always assumed. Then, it follows that t(e) = o(e), and o(e) = t(e).
A loop edge e ∈ E is such that o(e) = t(e), and parallel edges e 1 , e 2 ∈ E are such that e 1 = e 2 with o(e 1 ) = o(e 2 ) and t(e 1 ) = t(e 2 ). Parallel edges are not assumed to exist in any graph in this paper. Then one defines the groups of 0-chains and 1-chains on a graph G with coefficients R respectively. The spaces of functions on C 0 (G, R) and C 1 (G, R) are denoted by Elements of the subset of C 0 (G, R) being dual to C 0 (G, R) are referred to as 0-cochains. Similarly, elements of the subset of C 1 (G, R) being dual to C 1 (G, R) are referred to as 1-cochains. The set C 0 (G, R) is also denoted by Λ 0 (G, R) in this paper. The subsets Λ 1 (G, R) ⊂ C 1 (G, R) and S 1 (G, R) ⊂ C 1 (G, R) are defined by where ω(e) is understood as ω(e) ∈ R. In what follows, C 0 (G, R) and C 1 (G, R) are often abbreviated as C 0 (G) and C 1 (G), respectively. Similar abbreviations will be adopted.
The boundary operator is defined as The dual of the boundary operator, the coboundary map, is defined by If f is a 0-cochain, then it follows that (df )(e) = f (∂e). For any loop edge e ∈ E, it follows from f (t(e)) = f (o(e)) that (df )(e) = 0. This df is indeed an element belonging to Λ 1 (G), as shown below. For any f ∈ Λ 0 (G) and e ∈ E, one has that To define inner products, one introduces some measures. Let m V and m E be elements of C 0 (G) and S 1 (G), such that This m E ∈ S 1 (G) is referred to as a reversible measure.
The following are inner products Associated with this set of inner products, one defines the co-derivative d † : where This operator d † is the adjoint one of d as shown below.
Remark 2.1. The corresponding operator analogous to − d † in the continuous standard Riemannian geometry is referred to as divergence. Thus, − d † can be referred to as divergence on graph [14]. This operator The Laplacian acting on Λ 0 (G), ∆ V : Λ 0 (G) → Λ 0 (G), is defined as The explicit form of its action is obtained as follows. By putting ω = df for f ∈ Λ 0 (G), one has This operator is self-adjoint as shown below.
Proof. It follows that This operator is self-adjoint as shown below.
Proof. It can be proven by straightforward calculations. To this end, we put f 1 = d † ω 1 ∈ Λ 0 (G) and then it follows that Most of the operators and their properties discussed so far are well-known [10]. On the other hand, those discussed in the next subsection are not standard ones.

Operators for master equations
To discuss master equations, one introduces The operator associated with ϕ ∈ C 1 R (G) is defined as Condition in (8) is to guarantee the property d The above equality is verified as Remark 2.2. In the case that ϕ is such that ϕ(e) = 1 for any e ∈ E, one has that d ϕ f = df for any f ∈ C 0 (G). The operator d ϕ is an analogue of the one introduced in Ref. [15].
As well as the case for , E , one introduces an inner product on C 1 A (G) , associated with a reversible measure m E ∈ S 1 (G). The value of the inner product is the same as (2): With this inner product, one introduces the co-derivative on Proof. Substituting , ω(e) = − ϕ( e ) ω( e ), and one has Although the following operators will not be used in Section 3, the Laplacian is discussed for the sake of completeness. The Laplacian ∆ ϕ V : C 0 (G) → C 0 (G) is defined as The explicit form of its action is obtained as follows. By putting ω = d ϕ f for f ∈ C 0 (G), one has This operator is self-adjoint :

Master equations
Master equations are written as where {x} are discrete states, {w x→y } a transition matrix that describes the transition rate from a state x to another y. The equilibrium state at x is denoted by p eq (x). In what follows, these objects, {x}, {w x→y }, and {p eq (x)}, are treated as given data.
In this section, a graph formulation of master equations is shown in terms of objects developed in Section 2. Main claims of this paper and their consequences are then provided.

Graph formulation
Introduce a graph G = (V, E) associated with the given data {x} and {w x→y } so that • w(e) = w x→y for e ∈ E such that t(e) = y and o(e) = x • e ∈ E for any e ∈ E. Also, if w(e) = w y→x does not exist, then let w be such that w(e) = 0.
Then, regard p and w as follows.
• p ∈ Λ 0 (G) so that p(x) ∈ R for any x ∈ V .
• w ∈ C 1 (G) so that w(e) ∈ R for any e ∈ E.
One verifies that I(e) = −I(e), and that I(e) is the summand in the right hand side of (11), I(e) = φ(e) − φ( e ) = − p( o(e) ) w(e) + p( o(e) ) w( e ) = − p( o(e) ) w(e) + p( t(e) ) w( e ), and then, d dt The physical meaning of the case of I(e) = 0 for a given e ∈ E is that the probability flow or current is locally balanced.
To describe some measures, introduce In terms of these objects, one has the following.
Then (11) is identical to Proof. With the choices m V and m E , the co-derivative d † defined in (3) reduces to the one such that From this and (11), one has d dt for any x ∈ V . Thus, the desired expression is obtained.
It is worth mentioning that (13) is written as a form of a continuity equation in continuum mechanics where div : Λ 1 (G) → Λ 0 (G) has been defined in (4). Theorem 3.1 deals with p ∈ Λ 0 (G). On the other hand, an equality on Λ 1 (G) associated with the master equations are obtained from (13) as d dt dp = − dd † I = ∆ E I, where ∆ E has been defined in (7).
Remark 3.1. Notice that the present formalism is extended to the time-discrete case. Such a discrete system is immediately obtained from (13) as where p in I has been identified as p t , and τ > 0 is a constant.

Expectation values
Much attention is devoted to expectation values with respect to p in applications of master equations. To discuss such expectation values, one introduces O 0 ∈ Λ 0 (G) that is to be summed over vertexes at fixed time. In this subsection, m V ∈ Λ 0 (G) and m E ∈ S 1 (G) are chosen as (12). The expectation value of O 0 with respect to p is denoted by where the notation p t is to emphasize the time dependence for p. From the master equations (13) and Remark 3.2. Combining the sum over vertexes at fixed time t, and (15), one verifies that the derivative of 1 V = E p [ 1 ] with respect to time vanishes: whereṗ denotes derivative of p with respect to time t,ṗ = dp/dt.
The inner product for O 1 A that does not depend on time t, one uses (14) to obtain

Detailed balance conditions
Let p eq be a probability distribution function so that x p eq (x) = 1.
Impose for any states x and y p eq (x) w x→y = p eq (y) w y→x , which are known as the detailed balance conditions. These conditions are written in the graph theoretic language as x∈V p eq (x) = 1, and p eq (o(e)) w(e) = p eq (t(e)) w( e ).
Also due to (16), the m E ∈ S 1 (G) such that has the property that m E (e) = m E ( e ). Thus, this m E ∈ S 1 (G) can be used for a reversible measure defining (2). The master equations (11) under the detailed balance conditions are written as This can be written in terms of d π , that is (9) with ϕ = π, as d dt The above equations involve d π . In (18), there is no loop edge contribution since π(e) = 1 and (d π p)(e) = 0 for any loop edge e ∈ E, (See Remark 2.2). Although there is no self-adjoint operator in (18), there exists a way to express master equations in terms of a Laplacian, which is accomplished by a change of variables.
The following is the main theorem in this paper.
respectively, where w(e) satisfies (16). Then introduce ψ t ∈ Λ 0 (G) such that This function ψ t satisfies the diffusion equations where ∆ V has been defined in (5).
Remark 3.5. Notice that the present formalism is extended to the time-discrete case. Such a discrete system is obtained as where τ > 0 is constant. To investigate the long-time behavior of the system that is (21), the following Lemmas will be used. Proof. For general ψ, it follows that The equality holds only when dψ = 0. It implies that Since ( dψ (0) )(e) = ψ (0) (t(e)) − ψ (0) (o(e)) = 0, ∀e ∈ E, and the assumption that the graph is connected, the solution is ψ (0) (x) = Ψ 0 with Ψ 0 being constant.
Proof. Introduce the dynamical system for ψ ∈ Λ 0 (G), where ψ = Ψ 0 1 V forms a fixed point set for (23). This F( ψ ) has the properties that due to Lemma 3.1, where 0 V is the element of Λ 0 (G) such that 0 V (x) = 0 for any x ∈ V . The later states that the fixed point set for (24) is expressed by ψ = 0 V . Then define L acting on Λ 0 (G) such that It immediately follows that L( ψ ) ≥ 0 for any t ∈ R. In addition, one has that dL( ψ)/dt ≤ 0 for any t ∈ R, due to From these properties, L( ψ) is a Lyapunov function. Applying these statements to the Lyapunov stability theorem, one has that ψ = 0 V in the dynamical system (24) is asymptotically stable. This yields that Ψ 0 1 V is asymptotically stable for (23).
Then one has the following.
Proof. It follows from Lemmas 3.1 and 3. On the other hand, the conservation law (22) holds. Thus, the left hand side of the equation above is unity : Therefore, lim It has been shown that discrete diffusion equations are derived from master equations under the conditions that the detailed balance conditions are satisfied. Its converse statement also holds.
Proposition 3.1. Let w ∈ C 1 (G) be a transition matrix, and p eq ∈ Λ 0 (G) equilibrium distribution function. Assume that the detailed balance conditions (16) hold. Choose m V ∈ Λ 0 (G) and m E ∈ S 1 (G) so that m V (x) = p eq (x) and m E (e) = w(e)p eq (o(e)) as in (19). Then, the diffusion equations (21) yields the master equations.
Proof. Introduce the inner products (1) and (2). Also, introduce p t ∈ Λ 0 (G) that depends on t ∈ R such that p t (x) = p eq (x)ψ t (x). Then multiplying both sides of the diffusion equations by p eq (x), one has p eq (x) d dt ψ t (x) = e∈E x p eq ( o(e) )w(e) [ ψ t ( t(e) ) − ψ t ( o(e) ) ] , ∀ x ∈ V.
This can also be written in terms of p t (x) = p eq (x)ψ t (x) as [ −p t ( o(e) )w(e) + p t ( t(e) )w(e) ] .
The following is an example of how to apply Theorem 3.2.
Example 3.1. Consider the master equations with states and transition rule prescribed by the following. Let N ≥ 2 be fixed, and i ∈ {1, . . . , N } with i + N = i, and p eq (i) = 1/N that does not depend on i. Also, let w i→j ∈ R >0 be where q is a constant satisfying 0 ≤ q ≤ 1/2. One can verify that this system satisfies the detailed balance conditions as p eq (i) w i→i±1 = p eq (i ± 1) w i±1→i , i ∈ {1, . . . , N }. By regarding this system as a graph, one observes that there are loop edges that correspond to the case i = j. However, by recalling Remark 3.6, one realizes that there is no loop edge contribution for the equations of ψ t (i). Theorem 3.2 then reads d dt ψ t (i) = q [ ψ t (i + 1) + ψ t (i − 1) − 2 ψ t (i) ] , i ∈ V, which are indeed the discrete diffusion equations.