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We derive a class of mesoscopic virial equations governing energy partition between conjugate position and momentum variables of individual degrees of freedom. They are shown to apply to a wide range of nonequilibrium steady states with…
We study a class of Hamilton-Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish…
The Hamiltonian dynamics of the classical $\phi^4$ model on a two-dimensional square lattice is investigated by means of numerical simulations. The macroscopic observables are computed as time averages. The results clearly reveal the…
Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system…
The ergodic control problem for a non-degenerate controlled diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton-Jacobi-Bellman (HJB) equation. A…
The variational method is very important in mathematical and theoretical physics because it allows us to describe the natural systems by physical quantities independently from the frame of reference used. A global and statistical approach…
Dynamic heterogeneity has often been modeled by assuming that a single-particle observable, fluctuating at a molecular scale, is influenced by its coupling to environmental variables fluctuating on a second, perhaps slower, time scale.…
We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for…
The ability to manipulate and control fluid flows is of great importance in many scientific and engineering applications. Here, a cluster-based control framework is proposed to determine optimal control laws with respect to a cost function…
Equilibrium statistical mechanics is intended to link the microscopic dynamics of particles to the thermodynamic laws for macroscopic quantities. However, the modern statistical theory is faced with significant difficulties, as applied to…
The stochastic differential equations for a model of dissipative particle dynamics, with both total energy and total momentum conservation at every time-step, are presented. The algorithm satisfies detailed balance as well as the…
Stochastic unravelings allow to efficiently simulate open system dynamics, yet their application has traditionally been restricted to master equations that preserve both Hermiticity and trace. In this work, we introduce a general framework…
We improve recently introduced consensus-based optimization method, proposed in [R. Pinnau, C. Totzeck, O. Tse and S. Martin, Math. Models Methods Appl. Sci., 27(01):183--204, 2017], which is a gradient-free optimization method for general…
This work introduces a non-intrusive model reduction approach for learning reduced models from partially observed state trajectories of high-dimensional dynamical systems. The proposed approach compensates for the loss of information due to…
Recovering dynamical equations from observed noisy data is the central challenge of system identification. We develop a statistical mechanics approach to analyze sparse equation discovery algorithms, which typically balance data fit and…
Recent developments in multiscale computation allow the solution of ``coarse equations'' for the expected macroscopic behavior of microscopically/stochastically evolving particle distributions without ever obtaining these coarse equations…
We present a new methodology for studying non-Hamiltonian nonlinear systems based on an information theoretic extension of a renormalization group technique using a modified maximum entropy principle. We obtain a rigorous dimensionally…
Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these…
In this work, we investigate a stochastic control framework for global optimization over both Euclidean spaces and the Wasserstein space of probability measures, where the objective function may be non-convex and/or non-differentiable. In…
We develop a non-parametric, semimartingale optimal transport, calibration methodology for local volatility models with stochastic interest rate. The method finds a fully calibrated model which is the closest, in a way that can be defined…