Related papers: Heat and entropy flows in Carnot groups
Dunkel and Hilbert, "Consistent thermostatistics forbids negative absolute temperatures," Nature Physics, {\bf 10}, 67 (2014), and Hilbert, H\"anggi, and Dunkel, "Thermodynamic laws in isolated systems," Phys. Rev. E {\bf 90}, 062116 (2014)…
$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static…
We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step…
We show that for any liquid or solid with strong correlation between its $NVT$ virial and potential-energy equilibrium fluctuations, the temperature is a product of a function of excess entropy per particle and a function of density,…
We study a flow of $G_2$ structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time…
Entropy increase is fundamentally related to the breaking of time-reversal symmetry. By adding the 'extra dimension' associated with thermodynamic forces, we extend that discrete symmetry to a continuous symmetry for the dynamical…
We introduce a class of flows on the Wasserstein space of probability measures with finite first moment on the Cartan-Hadamard Riemannian manifold of positive definite matrices, and consider the problem of differentiability of the…
We establish basic properties of the heat flow on entire holomorphic functions that have order at most 2. We then look specifically at the action of the heat flow on the Gaussian analytic function (GAF). We show that applying the heat flow…
The effects of a heat current and gravity in liquid 4He near the superfluid transition are investigated for temperatures above and below T_lambda. We present a renormalization-group calculation based on model F for the Green's function in a…
Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth H\"{o}rmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous…
We study the Fokker-Planck equation as the hydrodynamic limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the rate functional, that characterizes the large deviations from the…
In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of linear parabolic initial-and final boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary…
In this article we derive Harnack estimates for conjugate heat kernel in an abstract geometric flow. Our calculation involves a correction term D. When D is nonnegative, we are able to obtain a Harnack inequality. Our abstract formulation…
The thermodynamics of quantum systems coupled to periodically modulated heat baths and work reservoirs is developed. By identifying affinities and fluxes, the first and second law are formulated consistently. In the linear response regime,…
In this paper we provide another way to deduce the Loomis-Whitney inequality on higher dimensional Heisenberg groups $\mathbb{H}^n$ based on the one on the first Heisenberg group $\mathbb{H}^1$ and the known nonlinear Loomis-Whitney…
We develop a geometric theory of phase transitions (PTs) for Hamiltonian systems in the microcanonical ensemble. This theory allows to reformulate Bachmann's classification of PTs for finite-size systems in terms of geometric properties of…
Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various…
In this work, we study the global well-posedeness of the heat equation with variable time-dependent nonlinearity of the form $\varphi(t)f(u)$ on unimodular Lie groups when the differential operator arises as the sum of squares of…
We present an analysis on the convergence properties of the so-called geometric heat flow equation for computing geodesics (extremal curves) on Riemannian manifolds. Computing geodesics numerically in real time has become an important…
We study autocorrelation functions of energy, heat and entropy densities obtained by molecular dynamics simulations of supercritical Ar and compare them with the predictions of the hydrodynamic theory. It is shown that the predicted by the…