arXiv subjects

Jani Lukkarinen

Publications and source records attributed to Jani Lukkarinen.

46 records · Page 3Linked to original sources

arXiv2006

Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations

We study the so called crossing estimate for analytic dispersion relations of periodic lattice systems in dimensions three and higher. Under a certain regularity assumption on the behavior of the dispersion relation near its critical values, we prove that an analytic dispersion relation suppresses crossings if and only if it is not a constant on any affine hyperplane. In particular, this will then be true for any dispersion relation which is a Morse function. We also provide two examples of simple lattice systems whose dispersion relations do not suppress crossings in the present sense.

math-ph↗

arXiv2006

Energy Transport in Weakly Anharmonic Chains

We investigate the energy transport in a one-dimensional lattice of oscillators with a harmonic nearest neighbor coupling and a harmonic plus quartic on-site potential. As numerically observed for particular coupling parameters before, and confirmed by our study, such chains satisfy Fourier's law: a chain of length N coupled to thermal reservoirs at both ends has an average steady state energy current proportional to 1/N. On the theoretical level we employ the Peierls transport equation for phonons and note that beyond a mere exchange of labels it admits nondegenerate phonon collisions. These collisions are responsible for a finite heat conductivity. The predictions of kinetic theory are compared with molecular dynamics simulations. In the range of weak anharmonicity, respectively low temperatures, reasonable agreement is observed.

cond-mat.stat-mech↗

arXiv2005

Kinetic Limit for Wave Propagation in a Random Medium

We study crystal dynamics in the harmonic approximation. The atomic masses are weakly disordered, in the sense that their deviation from uniformity is of order epsilon^(1/2). The dispersion relation is assumed to be a Morse function and to suppress crossed recollisions. We then prove that in the limit epsilon to 0 the disorder averaged Wigner function on the kinetic scale, time and space of order epsilon^(-1), is governed by a linear Boltzmann equation.

math-ph↗

arXiv2003

Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the "exterior" left and right heat baths are at specified values T_L and T_R, respectively, while the temperatures of the "interior" baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as 1/(l_d*d) where l_d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.

math-ph↗

arXiv2000

On the use of non-canonical quantum statistics

We develop a method using a coarse graining of the energy fluctuations of an equilibrium quantum system which produces simple parameterizations for the behaviour of the system. As an application, we use these methods to gain more understanding on the standard Boltzmann-Gibbs statistics and on the recently developed Tsallis statistics. We conclude on a discussion of the role of entropy and the maximum entropy principle in thermodynamics.

cond-mat.stat-mech↗

arXiv2000

Lattice computation of energy moments in canonical and Gaussian quantum statistics

We derive a lattice approximation for a class of equilibrium quantum statistics describing the behaviour of any combination and number of bosonic and fermionic particles with any sufficiently binding potential. We then develop an intuitive Monte Carlo algorithm which can be used for the computation of expectation values in canonical and Gaussian ensembles and give lattice observables which will converge to the energy moments in the continuum limit. The focus of the discussion is two-fold: in the rigorous treatment of the continuum limit and in the physical meaning of the lattice approximation. In particular, it is shown how the concepts and intuition of classical physics can be applied in this sort of computation of quantum effects. We illustrate the use of the Monte Carlo methods by computing canonical energy moments and the Gaussian density of states for charged particles in a quadratic potential.

cond-mat.stat-mech↗

arXiv1999

Using lattice methods in non-canonical quantum statistics

We define a natural coarse-graining procedure which can be applied to any closed equilibrium quantum system described by a density matrix ensemble and we show how the coarse-graining leads to the Gaussian and canonical ensembles. After this motivation, we present two ways of evaluating the Gaussian expectation values with lattice simulations. The first one is computationally demanding but general, whereas the second employs only canonical expectation values but it is applicable only for systems which are almost thermodynamical.

hep-lat↗

arXiv1998

Derivation and Improvements of the Quantum Canonical Ensemble from a Regularized Microcanonical Ensemble

We develop a regularization of the quantum microcanonical ensemble, called a Gaussian ensemble, which can be used for derivation of the canonical ensemble from microcanonical principles. The derivation differs from the usual methods by giving an explanation for the, at the first sight unreasonable, effectiveness of the canonical ensemble when applied to certain small, isolated, systems. This method also allows a direct identification between the parameters of the microcanonical and the canonical ensemble and it yields simple indicators and rigorous bounds for the effectiveness of the approximation. Finally, we derive an asymptotic expansion of the microcanonical corrections to the canonical ensemble for those systems, which are near, but not quite, at the thermodynamical limit and show how and why the canonical ensemble can be applied also for systems with exponentially increasing density of states. The aim throughout the paper is to keep mathematical rigour intact while attempting to produce results both physically and practically interesting.

cond-mat.stat-mech↗

arXiv1997

Lattice Approximation of Quantum Statistical Traces at a Complex Temperature

We prove that the simple condition on the potential V, \int exp(-t V) < \infty for all t>0, is sufficient for the lattice approximation of the trace Tr[A exp(-b H)] with (Re b)>0 to work for all bounded functions A and a large class of potentials. As a by-product we obtain an explicit bound for the real-temperature lattice kernels.

hep-th↗

arXiv1997

Lattice Simulations of the Quantum Microcanonical Ensemble

We propose a method for the numerical computation of microcanonical expectation values--i.e. averages over energy eigenstates with the same eigenvalue-without any prior knowledge about the spectrum of the Hamiltonian. This is accomplished by first defining a new Gaussian ensemble to approximate the microcanonical one. We then develop a ``lattice theory'' for this Gaussian ensemble and propose a Monte Carlo integration of the expectation values of the lattice theory.

hep-th↗