arXiv subjects

Richard Kleeman

Publications and source records attributed to Richard Kleeman.

6 recordsLinked to original sources

arXiv2019

A non-equilibrium theoretical framework for statistical physics with application to turbulent systems and their predictability

A new theoretical approach to non-equilibrium statistical systems has recently been proposed by the author, a co-author and others. It is based on a variational principle which is associated with the discrepancy of a path through thermodynamical space to one following Liouvillean evolution. In this contribution the approach is extended in such a way that it can be applied to a wide range of practical non-equilibrium statistical systems such as those arising in turbulence but also to a general class of statistical physics models. The new methodology allows for application to autonomous dynamical systems generalizing the previous work which applied only to Hamiltonian systems. Furthermore it provides a general analysis of near equilibrium conditions which allows for a natural analysis of predictability limits in turbulent systems. Finally it describes a method is described for the numerical calculation of far from equilibrium thermodynamical trajectories.

cond-mat.stat-mech↗

arXiv2016

Near-Gaussian entropic functional calculation and density estimation using an asymptotic series

Near-Gaussian probability densities are common in many important physical applications. Here we develop an asymptotic expansion methodology for computing entropic functionals for such densities. The expansion proposed is a close relative of standard perturbation expansions in quantum field theory. We give novel results on the low-order effects of non-Gaussian even moments and asymmetry (e.g. skewness) on the entropy. The asymptotic expansion is also used to define a best fit maximum entropy density given a set of observed low order moments. The maximum entropy density estimation technique consists simply of the solution of a small set of algebraic equations and is therefore more straightforward numerically than classical maximum-entropy methods which rely on sophisticated convex optimization techniques.

math.ST↗

arXiv2016

Multitimescale method for approximating the path action relevant to non-equilibrium statistical physics

A path integral formalism has been proposed recently for non-equilibrium statistical physics applications by the author. In this contribution we outline an efficient method for its numerical evaluation. The method used is based on the multiscale MCMC method of Ceperley and co-workers in quantum applications. A significant new feature of the method proposed is that the time endpoint is not fixed and indeed the endpoint sample is the principle object of interest.

math.NA↗

arXiv2015

A path integral formalism for the closure of autonomous statistical systems

Recently a path integral formalism has been proposed by the author which gives the time evolution of moments of slow variables in a Hamiltonian statistical system. This closure relies on evaluating the informational discrepancy of a time sequence (path) of approximating densities from the Liouvillian evolution that an exact density must follow. The discrepancy is then used to weight all possible paths using a generalized Boltzmann principle. That formalism is extended here to deal with more general and realistic autonomous dynamical systems. There the divergence of the time derivative of dynamical variables need not vanish as it does in the Hamiltonian case and this property complicates the closure derivation. Many interesting and realistic applications are covered by this new formalism including those describing realistic turbulence and the relevant specifics of this situation are outlined. The practical issues associated with the implementation of the outlined formalism are also discussed.

math-ph↗

arXiv2014

A path integral formalism for non-equilibrium Hamiltonian statistical systems

A path integral formalism for non-equilibrium systems is proposed based on a manifold of quasi-equilibrium densities. A generalized Boltzmann principle is used to weight manifold paths with the exponential of minus the information discrepancy of a particular manifold path with respect to full Liouvillean evolution. The likelihood of a manifold member at a particular time is termed a consistency distribution and is analogous to a quantum wavefunction. The Lagrangian here is of modified generalized Onsager-Machlup form. For large times and long slow timescales the thermodynamics is of Oettinger form. The proposed path integral has connections with those occuring in the quantum theory of a particle in an external electromagnetic field. It is however entirely of a Wiener form and so practical to compute. Finally it is shown that providing certain reasonable conditions are met then there exists a unique steady-state consistency distribution.

math-ph↗

arXiv2013

A nonequilibrium statistical model of spectrally truncated Burgers-Hopf dynamics

Exact spectral truncations of the unforced, inviscid Burgers-Hopf equation are Hamiltonian systems with many degrees of freedom which exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers-Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m << n. In the reduced model the m-mode macrostates are associated with trial probability densities on the phase space of the n-mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best-fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton-Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behaviour is found to be well represented by the reduced equations.

math-ph↗