Improved diffusive approximation of Markov jump processes close to equilibrium

TL;DR

This work addresses the inadequacy of standard diffusive approximations for Markov jump processes in capturing large fluctuations, especially near and away from equilibrium. By introducing a diffusion tensor based on the logarithmic mean of forward and reverse jump rates, the authors derive a modified, non-linear Fokker–Planck equation that preserves the large-deviation structure of the original MJP to linear order in departures from detailed balance. In the scaling limit, they connect diffusion dynamics to macroscopic stochastic thermodynamics, proving that the improved diffusion reproduces steady-state fluctuations and near-equilibrium dynamics more accurately than the traditional Kramers–Moyal diffusion, including transient properties such as memory error rates. Through extensive CMOS circuit examples, they show reduced errors in steady states, improved eigenmode predictions, and correct metastable lifetimes, demonstrating practical impact for designing and analyzing stochastic electronic circuits in the presence of non-equilibrium fluctuations.

Abstract

Diffusive approximations of Markov jump processes often fail to accurately capture large fluctuations. This is confounding, as the rare events triggered by these large fluctuations, such as the failure of electronic memories, are often the object of interest. In this paper we present an improved diffusive approximation, extending a method previously limited to equilibrium systems. Using new tools from stochastic thermodynamics, we prove its validity to linear order in departures from equilibrium and demonstrate its superior accuracy over the Kramers-Moyal expansion in predicting both steady-state and transient properties, including the error rate of a non-equilibrium electronic memory.

Figures (8)

• Figure 1: Diffusive approximation of Markov Jump Processes. (a) The steady-state probability distribution $P_\text{ss}$ for a two-dimensional stochastic process with three metastable states ($A$, $B$, and $C$). The white line indicates a transition path from $A$ to $B$, driven by stochastic fluctuations. Inset: A "zoom-in" on the transition path reveals the underlying discrete Markov jump process. (b) A comparison of the effective energy landscape along the path from $A$ to $B$, given by the steady-state self-information $I_\text{ss}:=-\log P_\text{ss}$ for the exact process (yellow) and a typical diffusive approximation (blue). The following panels highlight common failure modes of standard diffusive approximations that our improved method is designed to address. Top panel: The approximation underestimates the barrier height between the two metastable states. Bottom panel: While standard diffusive approximations correctly capture local behavior around stable states, their inaccuracy for large fluctuations can lead to qualitatively wrong predictions, such as identifying the wrong most-stable state (the global maximum of $P_\text{ss}$) vellela2009.
• Figure 2: Stochastic CMOS circuits. (a) In the context of digital electronics, a logical NOT gate (left) is typically implemented as a CMOS inverter (right), which employs two field-effect transistors. In the examples that follow, $V_\text{in} = 0$ and $V_\text{dd}= -V_\text{ss} = V_T$, with $V_T \equiv k_BT/e$ the thermal voltage. (b) Each transistor is modeled in the subthreshold regime as a stochastic conduction channel between its source (S) and drain (D) terminals, with voltage-dependent Poisson rates $\lambda_{+/-}$, cf. freitasStochastic2021 for a detailed discussion. For the circuit examples used in this paper, $C_o=2C_T$ and $C_g = 4C_T$, where $C_T \equiv e/V_T$.
• Figure 3: Liouvillian eigenmodes of a CMOS inverter under different diffusive approximations. (a) Three slowest-decaying eigenvectors for the Liouvillian of the inverter circuit in Figure \ref{['fig:CMOS-explainer']}a, plotted together with the three slowest-decaying eigenmodes of the Liouvillian for both the Kramers-Moyal approximation and the improved diffusion approximation. (b) Representative mid--spectrum eigenmodes (11th, 16th, and 21st slowest) for the same system. (c) Fractional eigenmode error $v_2^\text{err}(x) \equiv v_2(x)-v_2^\text{exact}(x)$, for both the Kramers-Moyal and improved diffusive approximations. Parameters are the same as in panel (a).
• Figure 4: (a) Circuit diagram for a random-access memory (RAM) cell formed by two CMOS inverters connected in a feedback loop (left inset). The stochastic degrees of freedom here are the voltages $x_0,x_1$ at the two output terminals of the circuit. We consider $V_\text{dd} = -V_\text{ss}$ for the powering voltages. (b) Steady-state probability density for the circuit in panel (a), under the powering configuration $V_\text{dd} =-V_\text{ss} = 1.3V_T$, and with $\Omega = 11$. The bistable axis $x_1\equiv -x_0$ for the memory is marked with a dashed line. (c) Cut along the bistable axis of the large-deviation function $I_\text{ss}(\bm{x})$ for the circuit. Here, $V_\text{dd} = -V_\text{ss} =2.6V_T$. We show results obtained from both the Kramers-Moyal approximation as well as the improved diffusion approximation.
• Figure 5: Performance of improved diffusion approximation applied to a CMOS memory. (a) Error (as measured by the Kullback-Leibler divergence) in estimating the nonequilibrium steady-state of a CMOS random-access memory (RAM) cell (cf. Figure \ref{['fig:CMOS-pbit']}a), under the Kramers-Moyal and improved diffusion approximations, as a function of the powering voltage $V_\text{dd}$ for the circuit, in units of $V_T \equiv k_BT/e$. Here, $\Omega$ corresponds to the smallest eigenvalue of the Maxwell capacitance matrix of the underlying circuit. (b) Fractional error in $\lambda_1$, the first Liouvillian eigenvalue, for both approximations. At $V_\text{dd} = \log (2)\, V_T$ (vertical dashed line), the system undergoes a phase transition and becomes bistable.