Abstract
We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.
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Acknowledgements
The authors would like to thank Jim Portegies, Oliver Tse, Jasper Hoeksema, Georg Prokert, and Frank Redig for several interesting discussions and insightful remarks. This work was partially supported by NWO grant 613.009.101.
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Appendix A: Proof of Lemma 4.6
Appendix A: Proof of Lemma 4.6
The existence and uniqueness of weak solutions Xn follows from arguments similar to the proof of Lemma 8.3, and we omit this proof.
For the stochastic process Yn we prove the existence and uniqueness as follows. The main step is to show the unique existence of a fundamental solution of the parabolic partial differential equation \(\partial _{t} \rho - {\mathscr{L}}_{Y}^{*}\rho =0\) on Ωn:
Lemma A.1
Fix T > 0. For each y0 ∈Ωn there exists a unique fundamental solution (t,y)↦p(t,y; y0) of the equation \(\partial _{t} -\mathcal L_{Y}^{*} =0\) on Ωn with Neumann boundary conditions, i.e., a function \(p = p_{y_{0}}\in L^{1}((0,T)\times {\Omega }_{n})\) satisfying
for all \(\varphi \in C^{1,2}_{b}([0,T]\times {\Omega }_{n})\). In addition, for each t > 0 the function y↦p(t,⋅; y0) is non-negative and continuous, and satisfies \({\int \limits }_{{\Omega }_{n}} p(t,y;y_{0}) dy= 1\).
Informally, the fundamental solution p satisfies
Given this fundamental solution, the proof of Lemma 4.6 proceeds along classical lines. We construct a consistent family of finite-dimensional distributions \(P_{t_{1},\dots ,t_{k}}\in \mathcal {P}(({\Omega }_{n})^{k})\) in the usual way, by daisy-chaining copies of the fundamental solution p(tk − tk− 1,⋅; ⋅) (see e.g. [96, Th. 2.2.2]). By applying the maximum-principle method of [96, Cor. 3.1.3] we show that this consistent family satisfies the conditions of Kolmogorov’s continuity theorem, and Theorem 2.1.6 of [96] then implies that \(P_{t_{1},\dots ,t_{k}}\) is generated by a unique probability measure \(\mathbb P\) on the space \(C([0,\infty );{\Omega }_{n})\). This concludes the argument.
The main step therefore is the proof of Lemma A.1, which we now give.
Proof
Define for L > α the truncated state space
Fix an initial datum \(\phi \in C_{b}({{\Omega }_{n}^{L}})\), ϕ ≥ 0; by classical methods there exists a non-negative \(C^{1,2}_{b}\) solution u of the equation \(\partial _{t} u - \mathcal L_{Y}^{*} u = 0\) on \((0,T)\times {{\Omega }_{n}^{L}}\), ∂nu = 0 on \((0,\infty )\times {\partial {\Omega }_{n}^{L}}\), and u(t = 0) = ϕ. By integrating the equation over \([0,t]\times {{\Omega }_{n}^{L}}\) we find
We now take a sequence \(L\to \infty \) and choose ϕL ≥ 0 with \({\int \limits } \phi ^{L} = 1\) such that ϕL converges narrowly on Ωn to \(\delta _{y_{0}}\). For each T > 0, the corresponding solution (t,y)↦uL(t,y) is non-negative and has integral over [0,T] ×Ωn equal to T (where we extend uL on \({\Omega }_{n}\setminus {{\Omega }_{n}^{L}}\) by zero); by taking a subsequence we can therefore assume that uL converges weakly, in duality with Cc([0,T] ×Ωn), to a non-negative limit measure p with p([0,T] ×Ωn) ≤ T. By e.g. [18, Th. 6.4.1] the measure p has a continuous Lebesgue density on (0,T) ×Ωn, implying that for 0 < t < T we can write it as p(dtdt) = p(t,y)dtdy.
By e.g. [18, Th. 6.4.1] the measure p has a continuous Lebesgue density on (0,T) ×Ωn, implying that for 0 < t < T we can write it as p(dtdt) = p(t,y)dtdy. Since \({\int \limits }_{{\Omega }_{n}} u^{L}(t, y) dy =1\) for all t, and since narrow convergence implies narrow convergence of marginals, the measure p(⋅,Ωn) on [0,T] coincides with Lebesgue measure on [0,T], and therefore \({\int \limits }_{{\Omega }_{n}}p(t,y)dy = 1\) for all t ∈ (0,T).
We now show that the function p satisfies (1). Take a function \(\varphi \in C^{1,2}_{c}([0,T)\times {\Omega }_{n})\) satisfying ∂nφ = 0 on ∂Ωn. Then \(\varphi \in C^{1,2}_{c}([0,T)\times {{\Omega }_{n}^{L}})\) for sufficiently large L, and in the weak form of the equation \(\partial _{t} u^{L} - \mathcal L_{Y} u^{L} = 0\) with initial datum ϕ,
we can replace the domain of integration \({{\Omega }_{n}^{L}}\) by Ωn. By taking the limit \(L\to \infty \) we find for all such φ the property
By a standard approximation argument, using the fact that the total mass of p is finite, this identity can be shown to hold for all \(\varphi \in C^{1,2}_{b}([0,T]\times {\Omega }_{n})\) with φ(t = T) = 0. This proves (1). By taking φ in Eq. 1 to be a function only of t, we also find \(\partial _{t} {\int \limits }_{{\Omega }_{n}} p(t,y) dy = 0\) in distributional sense, and therefore p(t,⋅) has unit mass for all time t.
Finally, we prove the uniqueness of p, which also implies that the final time T can be taken equal to \(\infty \). Let p be a finite measure on [0,T] ×Ωn that satisfies the weak (1) with initial datum equal to zero, i.e. assume that for all \(\varphi \in C^{1,2}([0,T]\times {\Omega }_{n})\) with φ(t = T) = 0,
Fix χ ∈ Cb([0,T] ×Ωn). By arguments very similar to those above we can find a solution \(\varphi \in C^{1,2}([0,T]\times {\Omega }_{n})\) of the equation
By substituting this φ in Eq. 5 we find
This implies that p is the zero measure on [0,T] ×Ωn, and proves the uniqueness of solutions of Eq. 1. □
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Peletier, M., Gavish, N. & Nyquist, P. Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System. Potential Anal 58, 71–121 (2023). https://doi.org/10.1007/s11118-021-09933-0
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DOI: https://doi.org/10.1007/s11118-021-09933-0