Related papers: A complete answer to the Gaveau--Brockett problem
We show how using a special relativistic kinetic equation with a BGK- like collision operator the ensuing expression for the heat flux can be casted in the form required by Classical Irreversible Thermodynamics. Indeed, it is linearly…
From the thermodynamic equilibrium properties of a two-level system with variable energy-level gap $\Delta$, and a careful distinction between the Gibbs relation $dE = T dS + (E/\Delta) d\Delta$ and the energy balance equation $dE = \delta…
Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation.…
In this paper we review the recently proposed path-integral counterpart of the Koopman-von Neumann operatorial approach to classical Hamiltonian mechanics. We identify in particular the geometrical variables entering this formulation and…
In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a…
This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic…
An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements…
We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator…
We develop a unified analytic treatment of the Horowitz--Polchinski string/black hole correspondence that systematically incorporates higher-derivative corrections to gravity. Working in Euclidean signature -- where the Euclidean black hole…
In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics $\sigma_\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as…
A new approach to obtain the closure formulas for the von K\'arm\'an--Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics…
This paper explicitly constructs the complete set of optimal sub-Riemannian geodesics starting from a point for certain Carnot groups of step two. These are groups of dimension 2n+1 equipped with a left-invariant distribution of dimension…
The approximate Carath\'eodory theorem states that given a compact convex set $\mathcal{C}\subset\mathbb{R}^n$ and $p\in\left[2,+\infty\right[$, each point $x^*\in\mathcal{C}$ can be approximated to $\epsilon$-accuracy in the $\ell_p$-norm…
The derivation of the efficiency of Carnot cycle is usually done by calculating the heats involved in two isothermal processes and making use of the associated adiabatic relation for a given working substance's equation of state, usually…
A novel equation of heat conduction is derived with the help of a generalized entropy current and internal variables. The obtained system of constitutive relations is compatible with the momentum series expansion of the kinetic theory. The…
In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and…
We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using the Method of Fokas or Unified Transform Method, we derive solution representations as the limit of solutions of…
A long standing open problem whether a heat engine with finite power achieves the Carnot efficiency is investigated. We rigorously prove a general trade-off inequality on thermodynamic efficiency and time interval of a cyclic process with…
We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $\mu$ and the heat conductivity $\kappa$ depend on the specific volume $v$ and the temperature…
For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the…