Action functional and quasi‐potential for the burgers equation in a bounded interval

Bertini, Lorenzo; Sole, Alberto De; Gabrielli, Davide; Jona-Lasinio, G.; Landim, Cláudio

doi:10.1002/cpa.20357

Cited by 16 publications

(27 citation statements)

References 39 publications

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“…We shall show that for E ≫ E 0 this model provides an example of a Lagrangian phase transition, see Section IV.D. This appears to be the first concrete example where this can be rigorously proven (Bertini et al, 2011).…”

Section: G An Example Of Lagrangian Phase Transitionmentioning

confidence: 77%

Macroscopic fluctuation theory

et al. 2015

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Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. We give a detailed review of this theory including its main predictions and most relevant applications.

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Section: G An Example Of Lagrangian Phase Transitionmentioning

confidence: 77%

Macroscopic fluctuation theory

et al. 2015

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“…Suppose that Assumption 2.1 holds. Given a time interval [s, t] ⊂ R, an initial state z ∈ X and a control u ∈ L 2 (s, t; U ) we consider the state equation (5) and its mild solution y(·; s, x, u), given by (7). We define the class of controls u(·) bringing the state y(·) from a fixed z ∈ X at time s to a given target x ∈ X at time t:…”

Section: General Formulationmentioning

confidence: 99%

Minimum energy for linear systems with finite horizon: a non-standard Riccati equation

Acquistapace

Gozzi

2017

Math. Control Signals Syst.

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This paper deals with a non-standard infinite dimensional linear-quadratic control problem arising in the physics of non-stationary states (see e.g. [6]): finding the minimum energy to drive a fixed stationary statex = 0 into an arbitrary non-stationary state x. The Riccati Equation (RE) associated to this problem is not standard since the sign of the linear part is opposite to the usual one, thus preventing the use of the known theory.Here we consider the finite horizon case. We prove that the linear selfadjoint operator P (t), associated to the value function, solves the above mentioned RE (Theorem 4.12). Uniqueness does not hold in general but we are able to prove a partial uniqueness result in the class of invertible operators (Theorem 4.13). In the special case where the involved operators commute, a more detailed analysis of the set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples of applications are given.

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“…In the case of inhomogeneous boundary data u − = u + , the quasi-potential V a,b is in general a nonlocal functional and, as its definition involves the solution of a difficult dynamical problem, its direct analysis does not appear feasible. For the specific case of the Burgers equation here considered and the choice σ(v) = v(1−v), in [4] it is shown that the quasi-potential V a,b can be expressed in terms of a much simpler variational problem which requires to optimize over functions of a single variable rather than on all paths as in (1.4).…”

Section: Introductionmentioning

confidence: 99%

A Variational Approach to the Stationary Solutions of the Burgers Equation

Bertini¹,

Ponsiglione²

2012

SIAM J. Math. Anal.

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Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced in [4], we analyze a Lyapunov functional for such equation which gives the large deviations asymptotics of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. We focus on boundary data corresponding to a standing wave solution to the Burgers equation in the whole line. In this case, the limiting functional has in fact a one-parameter family of minimizers and we analyze the so-called development by Γ-convergence; this amounts to compute the sharp asymptotic cost corresponding to a given shift of the stationary solution.2000 Mathematics Subject Classification. Primary 35A15, 34B40; Secondary 35B40, 82B24.

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Action functional and quasi‐potential for the burgers equation in a bounded interval

Cited by 16 publications

References 39 publications

Macroscopic fluctuation theory

Macroscopic fluctuation theory

Minimum energy for linear systems with finite horizon: a non-standard Riccati equation

A Variational Approach to the Stationary Solutions of the Burgers Equation

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