Potential in stochastic differential equations: novel construction

There is a whole range of emergent phenomena in a complex network such as robustness, adaptiveness, multiple-equilibrium, hysteresis, oscillation and feedback. Those non-equilibrium behaviours can often be described by a set of stochastic differential equations. One persistent important question is the existence of a potential function. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. We present an explicit construction procedure to obtain the potential and relevant quantities. In the procedure no reference to the Fokker–Planck equation is needed. The availability of the potential suggests that powerful statistical mechanics tools can be used in nonequilibrium situations. PACS number: 02.50.Ey Let us consider an n component network whose dynamics are described by a set of stochastic differential equations [1]. q̇tj = fj (qt ) + ζj (qt , t). (1) The question is whether or not we can find a potential function from equation (1) which gives a global description of dynamics. Here q̇tj = dqtj /dt with j = 1, 2, . . . , n and the subscript t for q indicates that the state variable q is a function of time. The value of jth component is denoted by qj . The network state variable forms an n dimensional vector q = (q1, q2, . . . , qn) in the state space. Here the superscript τ denotes the transpose. The state variable may be the values of particle coordinates in physics or the protein numbers in a signal transduction pathway or any other possible quantities specifying the network. Let fj (q) be the deterministic nonlinear force on the jth component, which includes both the effects from other components and itself, and ζj (q, t) the random force. For simplicity we will assume that fj is a smooth function explicitly independent of time. To be specific, the noise will be assumed to be Gaussian and white with the variance, 〈ζi(qt , t)ζj (qt ′ , t ′)〉 = 2Dij (qt )δ(t − t ′) (2) 0305-4470/04/030025+06$30.00 © 2004 IOP Publishing Ltd Printed in the UK L25 L26 Letter to the Editor and zero mean, 〈ζj 〉 = 0. Here δ(t) is the Dirac delta function and 〈· · ·〉 indicates the average with respect to the dynamics of the stochastic force. By the physics and chemistry convention the semi-positive definite symmetric matrix D = {Dij } with i, j = 1, 2, . . . , n is the diffusion matrix. Equation (2) also implies that, in situations where the temperature T can be defined, we have set kBT = 1 with kB the Boltzmann constant. We remark that if an average over the stochastic force ζ , a Wiener noise, is performed, equation (1) is reduced to the following equation in dynamical systems: 〈q̇t 〉 = 〈f(qt )〉 = f(〈qt 〉). The last equality is due to the fact that at same time t, the noise and the state variables are independent of each other. It is equivalent to the fact that the noise can be switched out without affecting the deterministic force, a process demonstrated possible in physics in dealing with environmental effects [2]. A broad range of phenomena in both natural and social sciences has been described by such a deterministic equation [3]. Because of its importance and usefulness, repeated attempts have been made to construct a potential function [4–7]. The effort had, however, only limited success [8]. The usefulness of a potential reemerges in the current study of dynamics of gene regulatory networks [9, 10], which would again require its construction in complex network dynamics. It has been observed that the nonlinear dynamics is in general dissipative (tr(F ) = 0), asymmetric (Fji = Fij ), and stochastic (ζ = 0). Here, the force matrix F is defined as Fij = ∂fi/∂qj i, j = 1, . . . , n. (3) and the trace is equal to the divergence of the force: tr(F ) = ∂ · f = nj=1 ∂fj/∂qj . The combination of those three features prevents any direct application of the insight from Hamiltonian dynamics and has been the main obstacle preventing the potential construction. In fact, the asymmetry of dynamics has been characterized as the hallmark of the network in a state far from thermal equilibrium, and has been proclaimed that it makes the usual theoretical approach near thermal equilibrium unworkable [4]. It is the goal of this letter to report that we have, nevertheless, discovered a novel construction that can take care of those dynamical features and can give us a potential function. We state, the explicit construction will be given below, that there exists a unique decomposition such that equation (1) can be rewritten in the following form: [S(qt ) + A(qt )]q̇t = −∂φ(qt ) + ξ(qt , t) (4) with the semi-positive definite symmetric matrix S(qt ), the anti-symmetric matrix A(qt ), the single-valued scalar function φ(qt ), and the stochastic force ξ(qt , t). Here ∂ is the gradient operator in the state variable space. It is straightforward to verify that the semi-positive definite symmetric matrix term is ‘dissipative’: q̇τt S(qt )q̇t 0; the anti-symmetric part does no ‘work’: q̇τt A(qt )q̇t = 0, therefore non-dissipative. Hence, it is natural to identify that the dissipation is represented by the semi-positive definite symmetric matrix S(q), the friction matrix, and the transverse force by the anti-symmetric matrix A(q), the transverse matrix. The scalar function φ(q) then acquires the meaning of potential energy. The decomposition from equation (1) to (4) may be called the φ-decomposition. However, without further constraint, equation (4) would be not unique. This may be illustrated by a simple counting. There are four apparent independent quantities in equation (4), while there are only two in equation (1). In order to have a unique form for equation (4), we may choose to impose the constraint on the stochastic force and the semi-positive definite symmetric matrix in the following manner: 〈ξ(qt , t)ξ τ (qt ′ , t ′)〉 = 2S(qt )δ(t − t ′) (5) Letter to the Editor L27 and 〈ξ(qt , t)〉 = 0. We observe that this constraint is consistent with the Gaussian and white noise assumption for ζ in equation (1). It may be called the stochasticity-dissipation relation. We further observe that the forms of equation (4) and (5) strongly resemble those of dissipative dynamics in quantum mechanics when both dissipative and Berry phase exists [11, 2]. The constrained φ-decomposition will be called the gauged φ-decomposition, which is indeed unique, as we will now demonstrate. We prove the existence and uniqueness of the gauged φdecomposition from equation (1) to (4) by an explicit construction. Using equation (1) to eliminate the velocity q̇t in equation (4), we have [S(qt ) + A(qt )][f(qt ) + ζ(qt , t)] = −∂φ(qt ) + ξ(qt , t). Noticing that the dynamics of noise is independent of that of the state variables we require that both the deterministic force and the noise satisfying following two equations separately. For the deterministic force, this leads to [S(q) + A(q)]f(q) = −∂φ(q) (6) suggesting a ‘rotation’ from the force f to the gradient of the potential φ at each point in the state space. We have dropped the subscript t. For stochastic force, we have: [S(q) + A(q)]ζ(q, t) = ξ(q, t) (7) which shows the same ‘rotation’ between the stochastic forces. Here we have also dropped the subscript t for the state variable. Using equation (2) and (5), equation (7) implies [S(q) + A(q)]D(q)[S(q)− A(q)] = S(q) (8) which suggests a duality between equation (1) and (4): a large friction matrix implies a small diffusion matrix. It is a generalization of the Einstein relation [12] to non-zero transverse matrix A. Next we introduce an auxiliary matrix function G(q) = [S(q) + A(q)]−1. (9) Here, the inversion ‘−1’ is with respect to the matrix. Using the property of the potential function φ: ∂ × ∂φ = 0((∂ × ∂φ)ij = (∂i∂j − ∂j ∂i)φ), equation (6) leads to ∂ × [G−1f(q)] = 0 (10) which gives n(n− 1)/2 conditions to determine the n×n auxiliary matrix G. The generalized Einstein relation, equation (8), leads to the following equation