Path Integral Derivation and Numerical Computation of Large Deviation Prefactors for Non-equilibrium Dynamics Through Matrix Riccati Equations

For many non-equilibrium dynamics driven by small noise, in physics, chemistry, biology, or economy, rare events do matter. Large deviation theory then explains that the leading order term of the main statistical quantities have an exponential behavior. The exponential rate is often obtained as the infimum of an action, which is minimized along an instanton. In this paper, we consider the computation of the next order sub-exponential prefactors, which are crucial for a large number of applications. Following a path integral approach, we derive the dynamics of the Gaussian fluctuations around the instanton and compute from it the sub-exponential prefactors. As might be expected, the formalism leads to the computation of functional determinants and matrix Riccati equations. By contrast with the cases of equilibrium dynamics with detailed balance or generalized detailed balance, we stress the specific non locality of the solutions of the Riccati equation: the prefactors depend on fluctuations all along the instanton and not just at its starting and ending points. We explain how to numerically compute the prefactors. The case of statistically stationary quantities requires considerations of non trivial initial conditions for the matrix Riccati equation.

This theoretical work uses no external dataset.

1. Thermodynamic potentials are usually defined as static properties, independently of the dynamics, but they also appear as quasipotential in effective dynamical theory, a classical example being macroscopic fluctuation theory [3].

We would like to thank A. Alfonsi, A. Levitt, G. Stoltz for useful comments on the numerical integration of matrix Riccati equations. We also thank two anonymous referees for their careful reading of the paper which helped improving the presentation of our results. The research leading to these results has received funding from the European Research Council under the European Union’s seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 616811). During the last stage of this work, F. Bouchet received support by a subagreement from the Johns Hopkins University with funds provided by Grant No. 663054 from Simons Foundation. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of Simons Foundation or the Johns Hopkins University. J. Reygner is supported by the French National Research Agency (ANR) under the programs EFI (ANR-17-CE40-0030) and QuAMProcs (ANR-19-CE40-0010).

Authors and Affiliations

1. Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France

   Freddy Bouchet

2. CERMICS, Ecole des Ponts, Marne-la-Vallée, France

   Julien Reygner

Corresponding author

Correspondence to Freddy Bouchet.

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose, have no competing interests to declare that are relevant to the content of this article. The authors declare no conflict of interest relevant to the content of this article.

Communicated by Jorge Kurchan.

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Appendix

The Appendix is organized as follows. In Sect. 1, the notion of time ordered exponential is introduced and a few properties are stated. Section 2 is dedicated to the resolution of the backward matrix Riccati equation appearing in (35). Finally, Sect. 3 presents the proof of (38).

Throughout the Appendix, we work under Assumptions (A1–4). Furthermore, as is argued in the beginning of Sect. 3, our overall purpose here is to emphasize the connections between fluctuation paths, large deviation prefactors, functional Gaussian determinants and matrix Riccati equations. Therefore, we chose not to obscure the exposition of our arguments with the exhaustive mathematical justification of technical details, which can however be checked on a case-by-case basis.

1.1 Time Ordered Exponentials

Throughout this section we let \(A=(A_t)_{t \le 0}\) be a bounded family of matrices of size \(d \times d\). The choice of \((-\infty ,0]\) as the set of times is convenient for our purpose but the contents of this section could easily be adapted to any interval.

For all \(t_0, t \le 0\), we denote by

the solution to the (two-sided) Cauchy problem

It is called the time ordered exponential of A from \(t_0\) to t. It is related with the notion of path ordering which is in particular used in quantum field theory. However, in the context of the present article, we are only dealing with bounded, finite-dimensional matrices, therefore the following properties, which will be used in the sequel, are elementary consequences of the Cauchy–Lipschitz theorem for linear ordinary differential equations.

1. (i)

   For all \(t_0, t_1, t_2 \le 0\), \(\lfloor \exp (\int _{s=t_0}^{t_2} A_s \mathrm {d}s)\rfloor = \lfloor \exp (\int _{s=t_1}^{t_2} A_s \mathrm {d}s)\rfloor \lfloor \exp (\int _{s=t_0}^{t_1} A_s \mathrm {d}s)\rfloor \).

2. (ii)

   For all \(t_0, t_1 \le 0\), \(\lfloor \exp (\int _{s=t_0}^{t_1} A_s \mathrm {d}s)\rfloor \) is invertible and \(\lfloor \exp (\int _{s=t_0}^{t_1} A_s \mathrm {d}s)\rfloor ^{-1} = \lfloor \exp (\int _{s=t_1}^{t_0} A_s \mathrm {d}s)\rfloor \).

3. (iii)

   For all \(t_0, t \le 0\), \(\det \lfloor \exp (\int _{s=t_0}^t A_s \mathrm {d}s)\rfloor = \det (\exp (\int _{s=t_0}^t A_s \mathrm {d}s)) = \exp (\int _{s=t_0}^t \text {tr}A_s \mathrm {d}s)\). As a consequence, if \((M_t)_{t \le 0}\) solves the matrix ordinary differential equation \(\dot{M}_t = A_t M_t\), then \(m_t = \det M_t\) satisfies \(\dot{m}_t = m_t \text {tr}A_t\).

1.2 Solving the Backward Matrix Riccati Equation

In this section, we solve the backward matrix Riccati equation which corresponds to the second line of (35). In order to alleviate the notations, we drop the superscript notation x on the quantities \(\varphi ^x\), \(Q^x_s\), \(R^x_s\), etc., therefore we are led to consider the backward Cauchy problem

We employ a quadrature method. Let us first define

By Proposition 1 and Remark 6, we have

in other words, \(K^0_t\) is a particular solution to the backward Riccati equation (110), but with a different behavior when \(t \uparrow 0\).

For all \(t \le 0\), we now let

and introduce

Notice that since \(\nabla V\) and \(\ell \) are assumed to be smooth, and by definition, the fluctuation path \(\varphi ^x\) is bounded in \(\mathbb {R}^d\), the family of matrices \((A_t)_{t \le 0}\) is bounded and therefore matches the setting of Sect. 1. Then \(Z_t\) solves the time-dependent Lyapunov equation

so that

We may now define

for all \(t<0\), and check that \(\dot{K}_t = K_t^2 + Q_t^\top K_t + K_t Q_t - 2R_t\) using (111), (112), (113) and (116). Furthermore, \(K^0_t\) remains bounded when \(t \uparrow 0\) whereas \(|t|Z_t^{-1}\) converges to \(I_d\), which implies that \(K_t^{-1}\) converges to 0.

As a conclusion, \(K_t\) is the solution to (110), which yields the formula (36) in Sect. 3.2.

Remark 8

It follows from the expression of \(K_t\) that \(|t|K_t\) converges to \(I_d\) when \(t \uparrow 0\). This implies that \(|t|^d\det K_t\) converges to 1, so that the third condition in (34) takes the more explicit form

1.3 Proof of (38)

In this section we prove the identity (38). The solution \((K_t)_{t<0}\) to the backward matrix Riccati equation was constructed in the previous section, and it is easily observed that \((\eta _t)_{t<0}\) is defined up to a multiplicative constant by

The appropriate multiplicative constant shall be chosen in accordance with the third condition of (34), which shall then provide the correct \(t \rightarrow -\infty \) limit for \(\eta _t\).

We first introduce the notation

for all \(t \le 0\).

Lemma 1

For all \(t_1, t_2 < 0\),

Proof

We first inject the formula (117) for \(K_s\) into (119) and obtain

By Property (i) of time ordered exponentials, for all \(t<0\),

so that

where we have used Property (ii) of time ordered exponentials. On the other hand,

therefore

and by Property (iii) of time ordered exponentials, \(m_t = \det U_t\) satisfies

which reduces to

thanks to Property (ii) of time ordered exponentials again. As a consequence,

which completes the proof.

In the next lemma we describe the \(t_2 \uparrow 0\) limit of the ratio \(\eta _{t_2}/\eta _{t_1}\).

Lemma 2

For all \(t_1 < 0\),

Proof

We fix \(t_1<0\) and let \(t_2\) grow to 0 in (121). By the definition of \(U_t\), \(|t_2|^{-1}U_{t_2}\) converges to \(I_d\), so that by Remark 8,

which completes the proof.

We finally address the \(t_1 \rightarrow -\infty \) limit of the expression obtained for \(\eta _{t_1}\).

Lemma 3

We have

Proof

Let \(t_1<0\). Using Properties (i) and (iii) of time ordered exponentials and (123), we rewrite

We are therefore led to compute the \(t_1 \rightarrow -\infty \) limit of \(Z_{t_1}\). To this aim we recall that \((Z_t)_{t \le 0}\) solves the time-dependent Lyapunov equation (115). In addition to the notation \(\bar{H} = \nabla ^2 V(\bar{x})\) introduced in Sect. 5.1, let us denote \(\bar{D} = \nabla \ell (\bar{x})\), so that taking the \(t \rightarrow -\infty \) limit of (115) shows that

satisfies the stationary Lyapunov equation

In addition,

• evaluating the identity (12) at \(\bar{x}\) yields \(\bar{D}^\top \bar{H} + \bar{H}\bar{D} = 0\),

• \(\bar{H}\) is assumed to be positive-definite.

As a consequence,

from which we deduce that

and the proof is completed.

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