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Generalized Langevin Equations for Systems with Local Interactions

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Abstract

We present a new method to approximate the Mori–Zwanzig (MZ) memory integral in generalized Langevin equations describing the evolution of smooth observables in high-dimensional nonlinear systems with local interactions. Building upon the Faber operator series we recently developed for the orthogonal dynamics propagator, and an exact combinatorial algorithm that allows us to compute memory kernels from first principles, we demonstrate that the proposed method is effective in computing auto-correlation functions, intermediate scattering functions and other important statistical properties of the observable. We also develop a new stochastic process representation of the MZ fluctuation term for systems in statistical equilibrium. Numerical applications are presented for the Fermi–Pasta–Ulam model, and for random wave propagation in homegeneous media.

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Notes

  1. Note that the projected fluctuation term \(\mathcal {P}e^{t\mathcal {QL}}\mathcal {QL}\varvec{u}(0)\) is identically zero since \(\mathcal {P}\mathcal {Q}=0\).

  2. Note that the ith component of the system (12) can be explicitly written as

    figure a
  3. The series expansion (19) needs to be handled with care if \(\mathcal {L}\) is an unbounded operator, e.g., the generator of the Koopman semigroup (3) (see [27], p. 481). In this case, \(e^{t\mathcal {L}}\) should be properly defined as

    $$\begin{aligned} e^{t\mathcal {L}}=\lim _{q\rightarrow \infty }\left( 1-\frac{t\mathcal {L}}{q}\right) ^{-q}. \end{aligned}$$

    In fact, \(\left( 1-t\mathcal {L}/q\right) ^{-1}\) is the resolvent of \(\mathcal {L}\) (modulus a constant factor), which can be rigorously defined for both bounded and unbounded operators.

  4. In our recent work [68] (§3.1) we proved that a Taylor series of the orthogonal dynamical propagation \(e^{t\mathcal {Q}\mathcal {L}}\) yields an expansion of the MZ memory integral that resembles the classical Dyson series in scattering theory.

  5. Note that \(\varvec{f}(t)\) is a random process obtained by mapping the random initial state \(\varvec{u}(0)=\varvec{u}(\varvec{x}_0)\) forward in time using the orthogonal dynamics propagator \(e^{t\mathcal {Q}\mathcal {L}(\varvec{x}_0)}\).

  6. The mean of \(u(t)=u(\varvec{x}(t,\varvec{x}_0))\) is necessarily time-independent at statistical equilibrium. In fact, at equilibrium we have that \(\varvec{x}_0\sim \rho _{eq}\) implies that \(\varvec{x}(t) \sim \rho _{eq}\) for all \(t\ge 0\). A statistically stationary process however, may not be stationary in phase space. Indeed, \(\varvec{x}(t)\) evolves in time, eventually in a chaotic way as it happens for systems with strange attractors and invariant measures.

  7. At statistical equilibrium the cross correlation functions are invariant under temporal shifts. This means that \(\langle u_{i}(s),u_{j}(t)\rangle _{eq}= \langle u_{i}(0),u_{j}(t-s)\rangle _{eq}\) for all \(t\ge s\). Hence, the solution to the projected MZ equation (16) is sufficient to compute the KL expansion of the multi-correlated process \(\varvec{u}(t)\), e.g., using the series expansion method proposed in [9].

  8. The partition function \(Z(\alpha ,\gamma )\) is defined as a functional integral over u(x) and p(x) (see, e.g., [58]).

  9. Note that for linear waves the wave momentum p(xt) is equal to \(\partial u(x,t)/\partial t\) (see Eq. (79)).

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Acknowledgements

This research was supported by the Air Force Office of Scientific Research (AFOSR) Grant FA9550-16-586-1-0092.

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Appendix A: Auto-correlation Function of Polynomial Observables

Appendix A: Auto-correlation Function of Polynomial Observables

In this Appendix we prove that the temporal auto-correlation function of phase space functions of the form \(u^n(t)=u^n(\varvec{x}(t,\varvec{x}_0))\), i.e.,

$$\begin{aligned} \langle u^{n}(0),u^{n}(t)\rangle _{\rho }\qquad n\in {\mathbb {N}} \end{aligned}$$
(A.1)

can be represented by replacing u(t) with the KL expansion (66), and then sending K to infinity. This result allows us to compute the auto-correlation function of \(u^n(t)\) based on the KL expansions of u(t).

Theorem A.1

Consider a zero-mean stationary stochastic process u(t), \(t\in [0,T]\), and assume that it has finite joint moments up to any desired order. Let

$$\begin{aligned} u_K(t) = \sum _{k=1}^K \sqrt{\lambda _k}\xi _k e_k(t), \qquad \end{aligned}$$
(A.2)

be the truncated Karhunen–Loève expansion of u(t). Then

$$\begin{aligned} \lim _{K\rightarrow \infty } \left| \langle u^{n}(0), u^{n}(t)\rangle _{\rho }- \langle u_{K}^{n}(0), u_{K}^{n}(t)\rangle _{\rho }\right| \qquad \forall n\in {\mathbb {N}}, \end{aligned}$$
(A.3)

i.e., \(\langle u_{K}^{n}(0), u_{K}^{n}(t)\rangle _{\rho }\) converges uniformly to \(\langle u^{n}(0), u^{n}(t)\rangle _{\rho }\) as \(K\rightarrow \infty \).

Proof

Let us define

$$\begin{aligned} \delta _K(t)&=\left| \langle u_{K}^{n}(t),u_{K}^{n}(0)\rangle - \langle u^{n}(t),u^{}(0)\rangle \right| \nonumber \\&=| \langle u_{K}^{n}(t),u_{K}^{n}(0)\rangle -\langle u^{n}(t),u^{n}_{K}(0)\rangle +\langle u^{n}(t),u^{n}_{K}(0)\rangle -\langle u^{n}(t),u^{n}(0)\rangle |\nonumber \\&= |\langle u_{K}^{n}(t)-u^{n}(t),u_{K}^{n}(0)\rangle +\langle u^{n}(t),u^{n}_{K}(0)-u^{n}(0)\rangle |\nonumber \\&\le |\langle u_{K}^{n}(t)-u^{n}(t),u_{K}^{n}(0)\rangle | +|\langle u^{n}(t),u^{n}_{K}(0)-u^{n}(0)\rangle |. \end{aligned}$$
(A.4)

The first term at the right hand side is of the form

$$\begin{aligned} a^{n}-b^{n}=(a-b)\sum _{i=0}^{n-1}a^{i}b^{n-1-i}. \end{aligned}$$

By using the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} |\langle u_{K}^{n}(t)-u^{n}(t),u_{K}^{n}(0)\rangle | =&\left| \langle (u_{K}(t)-u(t))\sum _{i=0}^{n-1}u_K^i(t) u^{n-1-i}(t),u_{K}^{n}(0)\rangle \right| \\ =&\left| \langle u_{K}(t)-u(t),u_{K}^{n}(0)\sum _{i=0}^{n-1} u_K^i(t)u^{n-1-i}(t)\rangle \right| \\ \le \,&\epsilon _K(t)\left\| u_{K}^{n}(0)\sum _{i=1}^{n-1} u_K^i(t)u^{n-1-i}(t)\right\| _{L^2}, \end{aligned}$$

where we defined \(\epsilon _K(t)=\Vert u_K(t)-u(t)\Vert _{L^2}\). It is well-known that \(\epsilon _K(t)\rightarrow 0\) as \(K\rightarrow \infty \) (see, e.g., [33]). By using the generalized Hölder’s inequality \(\Vert fg\Vert _{L^p}\le \Vert f\Vert _{L^{q}}\Vert g\Vert _{L^{q}}\), where \(2p=q\) and the Minkowski inequality, we obtain

$$\begin{aligned} |\langle u_{K}^{n}(t)-u^{n}(t),u_{K}^{n}(0)\rangle |&\le \epsilon _K(t)\Vert u_K^n(0)\Vert _{L^4}\sum _{i=1}^{n-1} \Vert u_K^i(t)u^{n-i-1}(t)\Vert _{L^4}\nonumber \\&\le \epsilon _K(t)\Vert u_K^n(0)\Vert _{L^4}\sum _{i=1}^{n-1} \Vert u_K^i(t)\Vert _{L^8}\Vert u^{n-i-1}(t)\Vert _{L^8}=C_1\epsilon _K(t), \end{aligned}$$
(A.5)

where

$$\begin{aligned} C_1 = \Vert u_K^n(0)\Vert _{L^4} \sum _{i=1}^{n-1}\Vert u_K^i(t)\Vert _{L^8}\Vert u^{n-i-1}(t)\Vert _{L^8} <\infty . \end{aligned}$$
(A.6)

Similarly, we have

$$\begin{aligned} |\langle u^{n}(t),u_{K}^{n}(0)-u^n(0)\rangle | \le \epsilon _K(0)\Vert u^n(0)\Vert _{L^4} \sum _{i=1}^{n-1}\Vert u_K^i(0)\Vert _{L^8}\Vert u^{n-i-1}(0)\Vert _{L^8} =C_2\epsilon _K(0). \end{aligned}$$
(A.7)

By combining (A.4), (A.5) and (A.7), we obtain

$$\begin{aligned} \lim _{K\rightarrow +\infty } \delta _K(t)\le \lim _{K\rightarrow +\infty } C_1\epsilon _K(t)+C_2\epsilon _K(0)=0, \end{aligned}$$

which proves the theorem. \(\square \)

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Zhu, Y., Venturi, D. Generalized Langevin Equations for Systems with Local Interactions. J Stat Phys 178, 1217–1247 (2020). https://doi.org/10.1007/s10955-020-02499-y

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