No Arabic abstract
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.
We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems. In particular we discuss the coupling of a quantum dot coupled to macroscopic charge carriers.
We consider the system of particles with equal charges and nearest neighbour Coulomb interaction on the interval. We study local properties of this system, in particular the distribution of distances between neighbouring charges. For zero temperature case there is sufficiently complete picture and we give a short review. For Gibbs distribution the situation is more difficult and we present two related results.
A coupled forward-backward stochastic differential system (FBSDS) is formulated in spaces of fields for the incompressible Navier-Stokes equation in the whole space. It is shown to have a unique local solution, and further if either the Reynolds number is small or the dimension of the forward stochastic differential equation is equal to two, it can be shown to have a unique global solution. These results are shown with probabilistic arguments to imply the known existence and uniqueness results for the Navier-Stokes equation, and thus provide probabilistic formulas to the latter. Related results and the maximum principle are also addressed for partial differential equations (PDEs) of Burgers type. Moreover, from truncating the time interval of the above FBSDS, approximate solution is derived for the Navier-Stokes equation by a new class of FBSDSs and their associated PDEs; our probabilistic formula is also bridged to the probabilistic Lagrangian representations for the velocity field, given by Constantin and Iyer (Commun. Pure Appl. Math. 61: 330--345, 2008) and Zhang (Probab. Theory Relat. Fields 148: 305--332, 2010) ; finally, the solution of the Navier-Stokes equation is shown to be a critical point of controlled forward-backward stochastic differential equations.
While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size $2n times 2n$ may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order $sqrt{log n}$. Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom-Rowlinson or Lenard-Jones-type interaction.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure could be taken as the error of adiabatic approximation. We prove under certain conditions, this error can be precisely estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example on which the applicability of the adiabatic theorem is questionable.