Abstract
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.
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Notes
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].
References
Abbot, D.S., Webber, R.J., Hadden, S., Weare, J.: Rare event sampling improves mercury instability statistics. arXiv:2106.09091 (2021)
Berglund, N.: Kramers’ law: validity, derivations and generalisations. Markov Process. Relat. Fields 19(3), 459–490 (2013)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory. Rev. Modern Phys. 87, 593–636 (2015)
Bouchet, F., Reygner, J.: Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes. Ann. Henri Poincaré 17(12), 3499–3532 (2016)
Bouchet, F., Nardini, C., Gawedzki, K.: Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example. J. Stat. Phys. 163, 1157–1210 (2016)
Bouchet, F., Rolland, J., Simonnet, E.: Rare event algorithm links transitions in turbulent flows with activated nucleations. Phys. Rev. Lett. 122(7), 074,502 (2019)
Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Euro. Math. Soc. 6(4), 399–424 (2004)
Callan, C.G., Jr., Coleman, S.R.: The fate of the false vacuum. 2. First quantum corrections. Phys. Rev. D 16, 1762–1768 (1977). https://doi.org/10.1103/PhysRevD.16.1762
Cohen, J.K., Lewis, R.M.: A ray method for the asymptotic solution of the diffusion equation. IMA J. Appl. Math. 3(3), 266–290 (1967)
Coleman, S.R.: The uses of instantons. Subnucl. Ser. 15, 805 (1979)
Dematteis, G., Grafke, T., Vanden-Eijnden, E.: Rogue waves and large deviations in deep sea. Proc. Natl. Acad. Sci. USA 115(5), 855–860 (2018)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, Stochastic Modelling and Applied Probability, vol. 38. Springer, Berlin (2010). (Corrected reprint of the second edition)
Dieci, L., Eirola, T.: Positive definiteness in the numerical solution of Riccati differential equations. Numer. Math. 67, 303–313 (1994)
Dieci, L., Eirola, T.: Preserving monotonicity in the numerical solution of Riccati differential equations. Numer. Math. 74, 35–47 (1996)
Dubois, F., Saïdi, A.: Unconditionnally stable scheme for Riccati equation. ESAIM Proc. 8, 39–52 (2000)
Ferré, G., Grafke, T.: Approximate optimal controls via instanton expansion for low temperature free energy computation. arXiv:2011.10990
Freidlin, M.I. and Wentzell, A.D.: Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften, vol. 260. Springer, Heidelberg (2012). Translated from the 1979 Russian original by Joseph Szücs. Third edition
Grafke, T., Vanden-Eijnden, E.: Numerical computation of rare events via large deviation theory. Chaos 29(6), 063118 (2019)
Grafke, T., Grauer, R., Schäfer, T.: Instanton filtering for the stochastic burgers equation. J. Phys. A 46(6), 062,002 (2013)
Grafke, T., Grauer, R., Schindel, S.: Efficient computation of instantons for multi-dimensional turbulent flows with large scale forcing. Commun. Comput. Phys. 18(3), 577–592 (2015)
Grafke, T., Schäfer, T., Vanden-Eijnden, E.: Sharp Asymptotic Estimates for Expectations, Probabilities, and Mean First Passage Times in Stochastic Systems with Small Noise. arXiv:2103.04837
Graham, R.: Macroscopic potentials, bifurcations and noise in dissipative systems. Noise Nonlinear Dyn. Syst. 1, 225–278 (1988)
Heymann, M., Vanden-Eijnden, E.: The geometric minimum action method: a least action principle on the space of curves. Commun. Pure Appl. Math. 61(8), 1052–1117 (2008)
Kampen, N.G.V.: Stochastic Processes in Physics and Chemistry. North-Holland Personal Library, 3rd edn. Elsevier, Amsterdam (2007)
Landim, C., Mariani, M., Seo, I.: Dirichlet’s and Thomson’s principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes. Arch. Ration. Mech. Anal. 231(2), 887–938 (2019)
Langer, J.S.: Theory of the condensation point. Ann. Phys. 41, 108–157 (1967). https://doi.org/10.1016/0003-4916(67)90200-X
Langer, J.: Excitation chains at the glass transition. Phys. Rev. Lett. 97(11), 115,704 (2006)
Laurie, J., Bouchet, F.: Computation of rare transitions in the barotropic quasi-geostrophic equations. New J. Phys. (2015). https://doi.org/10.1088/1367-2630/17/1/015009
Lee, J., Seo, I.: Non-reversible metastable diffusions with Gibbs invariant measure I: Eyring–Kramers formula. arXiv:2008.08291
Lu, Y., Stuart, A.M., Weber, H.: Gaussian approximations for transition paths in molecular dynamics. SIAM J. Math. Anal. 49(4), 3005–3047 (2017)
Ludwig, D.: Persistence of dynamical systems under random perturbations. SIAM Rev. 17(4), 605–640 (1975)
Maier, R.S., Stein, D.L.: Limiting exit location distributions in the stochastic exit problem. SIAM J. Appl. Math. 57(3), 752–790 (1997)
Paskal, N., Cameron, M.: An efficient jet marcher for computing the quasipotential for 2D SDEs. To appear in J. Sci. Comput
Ragone, F., Wouters, J., Bouchet, F.: Computation of extreme heat waves in climate models using a large deviation algorithm. Proc. Natl. Acad. Sci. U.S.A. 115(1), 24–29 (2018). https://doi.org/10.1073/pnas.1712645115
Sanz-Alonso, D., Stuart, A.M.: Gaussian approximations of small noise diffusions in Kullback–Leibler divergence. Commun. Math. Sci. 15(7), 2087–2097 (2017)
Schuss, Z.: Theory and Applications of Stochastic Processes: An Analytical Approach, Applied Mathematical Sciences, vol. 170. Springer, New York (2010)
Simonnet, E., Rolland, J., Bouchet, F.: Multistability and rare spontaneous transitions in barotropic \(\beta \)-plane turbulence. J. Atmos. Sci. 78(6), 1889–1911 (2021)
Vanden-Eijnden, E., Heymann, M.: The geometric minimum action method for computing minimum energy paths. J. Chem. Phys. 128(6), 61–103 (2008)
Woillez, E., Bouchet, F.: Instantons for the destabilization of the inner solar system. Phys. Rev. Lett. 125(2), 021,101 (2020)
Woillez, E., Zhao, Y., Kafri, Y., Lecomte, V., Tailleur, J.: Activated escape of a self-propelled particle from a metastable state. Phys. Rev. Lett. 122, 258,001 (2019). https://doi.org/10.1103/PhysRevLett.122.258001
Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Clarendon Press, Oxford (1996)
Acknowledgements
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).
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Appendix
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.
-
(i)
For all \(t_0, t_1, t_2 \le 0\), \(\lfloor \exp (\int _{s=t_0}^{t_2} A_s \mathrm {d}s)\rfloor = \lfloor \exp (\int _{s=t_1}^{t_2} A_s \mathrm {d}s)\rfloor \lfloor \exp (\int _{s=t_0}^{t_1} A_s \mathrm {d}s)\rfloor \).
-
(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 \(\lfloor \exp (\int _{s=t_0}^{t_1} A_s \mathrm {d}s)\rfloor ^{-1} = \lfloor \exp (\int _{s=t_1}^{t_0} A_s \mathrm {d}s)\rfloor \).
-
(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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Bouchet, F., Reygner, J. Path Integral Derivation and Numerical Computation of Large Deviation Prefactors for Non-equilibrium Dynamics Through Matrix Riccati Equations. J Stat Phys 189, 21 (2022). https://doi.org/10.1007/s10955-022-02983-7
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DOI: https://doi.org/10.1007/s10955-022-02983-7