arXiv subjects

Jan Naudts

Publications and source records attributed to Jan Naudts.

At least 37 records · Page 2Linked to original sources

arXiv2013

Quantum cooling by unitary transformations

We study the unitary time evolution of a simple quantum Hamiltonian describing a heat engine coupled to two heat baths. The engine is modeled as a three-level system. Each heat bath consists of a single harmonic oscillator. The engine is operated via time-dependent external fields. The S-matrix of the thermodynamic cycle is obtained in analytic form. We conjecture that the spectrum of this S-matrix contains a continuous part, and that this is a requirement for the operation as a heat engine. Energy currents flow in both directions through the engine. The balance of these currents determines whether the engine performs work or whether its operation requires the application of external forces.

math-ph↗

arXiv2013

Energy transfer using unitary transformations

We study the unitary time evolution of a simple quantum Hamiltonian describing two harmonic oscillators coupled via a three-level system. The latter acts as an engine transferring energy from one oscillator to the other and is driven in a cyclic manner by time-dependent external fields. The S-matrix of the cycle is obtained in analytic form. The total number of quanta contained in the system is a conserved quantity. As a consequence the spectrum of the S-matrix is purely discrete and the evolution of the system is quasi-periodic.

quant-ph↗

arXiv2013

The exponential family in abstract information theory

We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are those of information geometry. The context is general enough to include applications from outside statistics.

cs.IT↗

arXiv2011

Data Set Models and Exponential Families in Statistical Physics and Beyond

The exponential family of models is defined in a general setting, not relying on probability theory. Some results of information geometry are shown to remain valid. Exponential families both of classical and of quantum mechanical statistical physics fit into the new formalism. Other less obvious applications are predicted. For instance, quantum states can be modeled as points in a classical phase space and the resulting model belongs to the exponential family.

math-ph↗

arXiv2011

A multiplet analysis of spectra in the presence of broken symmetries

We introduce the notion of a generalised symmetry M of a hamiltonian H. It is a symmetry which has been broken in a very specific manner, involving ladder operators R and R*. In Theorem 1 these generalised symmetries are characterised in terms of repeated commutators of H with M. Breaking supersymmetry by adding a term linear in the supercharges is discussed as a motivating example. The complex parameter gamma which appears in the definition of a generalised symmetry is necessarily real when the spectrum of M is discrete. Theorem 2 shows that gamma must also be real when the spectrum of H is fully discrete and R and R* are bounded operators. Any generalised symmetry induces a partitioning of the spectrum of H in what we call M-multiplets. The hydrogen atom in the presence of a symmetry breaking external field is discussed as an example. The notion of stability of eigenvectors of H relative to the generalised symmetry M is discussed. A characterisation of stable eigenvectors is given in Theorem 3.

math-ph↗

arXiv2010

On the BCH formula of Rezek and Kosloff

The BCH formula of Rezek and Kosloff is a convenient tool to handle a family of density matrices, which occurs in the study of quantum heat engines. We prove the formula using a known argument from Lie theory.

math-ph↗

arXiv2010

On the thermodynamics of classical micro-canonical systems

We study the configurational probability distribution of a mono-atomic gas with a finite number of particles N in the micro-canonical ensemble. We give two arguments why the thermodynamic entropy of the configurational subsystem involves Renyi's entropy function rather than that of Tsallis. The first argument is that the temperature of the configurational subsystem is equal to that of the kinetic subsystem. The second argument is that the instability of the pendulum, which occurs for energies close to the rotation threshold, is correctly reproduced.

cond-mat.stat-mech↗

arXiv2010

A mechanism for pair formation in strongly correlated systems

We start from a Hamiltonian describing non-interacting fermions and add bosons to the model, with a Jaynes-Cummings-like interaction between the bosons and fermions. Because of the specific form of the interaction the model can be solved exactly. In the ground state, part of the electrons form bound pairs with opposite momentum and spin. The model also shows a gap in the kinetic energy of the fermions, but not in the spectrum of the full Hamiltonian. This gap is not of a mean-field nature, but is due to the Pauli exclusion principle.

cond-mat.str-el↗

arXiv2010

Electron pair forming in an integrable model

We consider a lattice model in which phonons scatter with pairs of electrons. All eigenvalues and eigenvectors can be obtained analytically. For a suitable choice of parameters the ground state consists of a Fermi sea of non-interacting electrons, with on top a layer of paired electrons. The binding energy of the electron pairs is partly cancelled by an increased kinetic energy. This results in a gap in the kinetic energy spectrum of the electrons.

cond-mat.str-el↗

arXiv2009

Analytic solutions for a three-level system in a time-dependent field

This paper generalizes some known solitary solutions of a time-dependent Hamiltonian in two ways: The time-dependent field can be an elliptic function, and the time evolution is obtained for a complete set of basis vectors. The latter makes it feasible to consider arbitrary initial conditions. The former makes it possible to observe a beating caused by the non-linearity of the driving field.

quant-ph↗

arXiv2009

The q-exponential family in statistical physics

The Boltzmann-Gibbs probability distribution, seen as a statistical model, belongs to the exponential family. Recently, the latter concept has been generalized. The q-exponential family has been shown to be relevant for the statistical description of small isolated systems. Two main applications are reviewed: 1. The distribution of the momentum of a single particle is a q-Gaussian, the distribution of its velocity is a deformed Maxwellian; 2. The configurational density distribution belongs to the q-exponential family. The definition of the temperature of small isolated systems is discussed. It depends on defining the thermodynamic entropy of a microcanonical ensemble in a suitable manner. The simple example of non-interacting harmonic oscillators shows that Renyi's entropy functional leads to acceptable results.

cond-mat.stat-mech↗

arXiv2009

Non-extensivity of the configurational density distribution in the classical microcanonical ensemble

We show that the configurational probability distribution of a classical gas always belongs to the q-exponential family. Hence, the configurational subsystem is non-extensive in the sense of Tsallis. One of the consequences of this observation is that the thermodynamics of the configurational subsystem is uniquely determined up to a scaling function. As an example we consider a system of non-interacting harmonic oscillators. In this example, the scaling function can be determined from the requirement that in the limit of large systems the microcanonical temperature of the configurational subsystem should coincide with that of the canonical ensemble. The result suggests that Renyi's entropy function is the relevant one rather than that of Tsallis.

cond-mat.stat-mech↗

arXiv2009

Analysis of the N=4 Hubbard ring using a counting operator

We prove three theorems about the use of a counting operator to study the spectrum of model Hamiltonians. We analytically calculate the eigenvalues of the Hubbard ring with four lattice positions and apply our theorems to describe the observed level crossings.

cond-mat.str-el↗

arXiv2009

Counting operator analysis of the discrete spectrum of some model Hamiltonians

The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R^\dagger which share some of the properties of creation and annihilation operators, and such that $M$ becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by the symmetries of H, but by those of a reference Hamiltonian H_ref, which is defined by H_ref=H-R-R^\dagger, and which commutes with M. Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.

cond-mat.str-el↗

arXiv2008

The quantum double well anharmonic oscillator in an external field

The aim of this paper is twofold. First of all, we study the behaviour of the lowest eigenvalues of the quantum anharmonic oscillator under influence of an external field. We try to understand this behaviour using perturbation theory and compare the results with numerical calculations. This brings us to the second aim of selecting the best method to carry out the numerical calculations accurately.

quant-ph↗

arXiv2008

Grand canonical ensemble in generalized thermostatistics

We study the grand-canonical ensemble with a fluctuating number of degrees of freedom in the context of generalized thermostatistics. Several choices of grand-canonical entropy functional are considered. The ideal gas is taken as an example.

cond-mat.stat-mech↗

arXiv2008

Transition records of stationary Markov chains

In any Markov chain with finite state space the distribution of transition records always belongs to the exponential family. This observation is used to prove a fluctuation theorem, and to show that the dynamical entropy of a stationary Markov chain is linear in the number of steps. Three applications are discussed. A known result about entropy production is reproduced. A thermodynamic relation is derived for equilibrium systems with Metropolis dynamics. Finally, a link is made with recent results concerning a one-dimensional polymer model.

cond-mat.stat-mech↗

arXiv2008

The 3-dimensional random walk with applications to overstretched DNA and the protein titin

We study the three-dimensional persistent random walk with drift. Then we develop a thermodynamic model that is based on this random walk without assuming the Boltzmann-Gibbs form for the equilibrium distribution. The simplicity of the model allows us to perform all calculations in closed form. We show that, despite its simplicity, the model can be used to describe different polymer stretching experiments. We study the reversible overstretching transition of DNA and the static force-extension relation of the protein titin.

cond-mat.stat-mech↗