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Jan Naudts

Publications and source records attributed to Jan Naudts.

At least 55 records · Page 3Linked to original sources
arXiv2008

Residual entropy in a model for the unfolding of single polymer chains

We study the unfolding of a single polymer chain due to an external force. We use a simplified model which allows to perform all calculations in closed form without assuming a Boltzmann-Gibbs form for the equilibrium distribution. Temperature is then defined by calculating the Legendre transform of the entropy under certain constraints. The application of the model is limited to flexible polymers. It exhibits a gradual transition from compact globule to rod. The boundary line between these two phases shows reentrant behavior. This behavior is explained by the presence of residual entropy.

cond-mat.stat-mech↗
arXiv2008

The q-exponential family in statistical physics

The notion of generalised exponential family is considered in the restricted context of nonextensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown that the configurational probability distributions of the microcanonical ensemble belong to the q-exponential family.

cond-mat.stat-mech↗
arXiv2008

Solving the von Neumann equation with time-dependent Hamiltonian. Part I: Method

The unitary operators U(t), describing the quantum time evolution of systems with a time-dependent Hamiltonian, can be constructed in an explicit manner using the method of time-dependent invariants. We clarify the role of Lie-algebraic techniques in this context and elaborate the theory for SU(2) and SU(1,1). We show that the constructions known as Magnus expansion and Wei-Norman expansion correspond with different representations of the rotation group. A simpler construction is obtained when representing rotations in terms of Euler angles. The many applications are postponed to Part II of the paper.

math-ph↗
arXiv2008

On the von Neumann equation with time-dependent Hamiltonian. Part II: Applications

This second part deals with applications of a general method to describe the quantum time evolution determined by a Schroedinger equation with time-dependent Hamiltonian. A new aspect of our approach is that we find all solutions starting from one special solution. The two main applications are reviewed, namely the Bloch equations and the harmonic oscillator with time-dependent frequency. Even in these well-known examples some new results are obtained.

math-ph↗
arXiv2008

Generalised exponential families and associated entropy functions

A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential families. The well-known result that the inequality of Cramer and Rao becomes an equality in the case of an exponential family can be generalised. However, this requires the introduction of escort probabilities.

math-ph↗
arXiv2007

Generalised thermostatistics using hyperensembles

The hyperensembles, introduced by Crooks in a context of non-equilibrium statistical physics, are considered here as a tool for systems in equilibrium. Simple examples like the ideal gas, the mean-field model, and the Ising interaction on small square lattices, are worked out to illustrate the concepts.

cond-mat.stat-mech↗
arXiv2007

Ensemble averaged entanglement of two-particle states in Fock space

Recent results, extending the Schmidt decomposition theorem to wavefunctions of identical particles, are reviewed. They are used to give a definition of reduced density operators in the case of two identical particles. Next, a method is discussed to calculate time averaged entanglement. It is applied to a pair of identical electrons in an otherwise empty band of the Hubbard model, and to a pair of bosons in the the Bose-Hubbard model with infinite range hopping. The effect of degeneracy of the spectrum of the Hamiltonian on the average entanglement is emphasised.

quant-ph↗
arXiv2006

The globule-coil transition in a mean field approach

In this paper we study the unfolding transition observed in polymer stretching experiments. We use a known model and extend it with a mean-field-like interaction. Expressions for all necessary thermodynamic variables as a function of the model parameters are derived in closed form. To obtain the exact force-extension relation at constant temperature, we invert these relations numerically. Below a critical temperature the model exhibits a sharp first order phase transition.

cond-mat.stat-mech↗
arXiv2006

Entanglement of a microcanonical ensemble

We replace time-averaged entanglement by ensemble-averaged entanglement and derive a simple expression for the latter. We show how to calculate the ensemble average for a two-spin system and for the Jaynes-Cummings model. In both cases the time-dependent entanglement is known as well so that one can verify that the time average coincides with the ensemble average.

quant-ph↗
arXiv2006

A generalized quantum microcanonical ensemble

We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by a combination of a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.

quant-ph↗
arXiv2006

Regularization as quantization in reducible representations of CCR

A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincaré covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to Rényi's $α$. It is shown that the ``Shannon limit" $α\to 1$ plays here a role of correspondence principle with the standard regularized formalism.

hep-th↗
arXiv2006

Special solutions of nonlinear von Neumann equations

We consider solutions of the non-linear von Neumann equation involving Jacobi's elliptic functions sn, cn, and dn, and 3 linearly independent operators. In two cases one can construct a state-dependent Hamiltonian which is such that the corresponding non-linear von Neumann equation is solved by the given density operator. We prove that in a certain context these two cases are the only possibilities to obtain special solutions of this kind. Well-known solutions of the reduced Maxwell-Bloch equations produce examples of each of the two cases. Also known solutions of the non-linear von Neumann equation in dimension 3 are reproduced by the present approach.

math-ph↗
arXiv2006

A two-parameter random walk with approximate exponential probability distribution

We study a non-Markovian random walk in dimension 1. It depends on two parameters eps_r and eps_l, the probabilities to go straight on when walking to the right, respectively to the left. The position x of the walk after n steps and the number of reversals of direction k are used to estimate eps_r and eps_l. We calculate the joint probability distribution p_n(x,k) in closed form and show that, approximately, it belongs to the exponential family.

math-ph↗
arXiv2005

Parameter estimation in nonextensive thermostatistics

Equilibrium statistical physics is considered from the point of view of statistical estimation theory. This involves the notions of statistical model, of estimators, and of exponential family. A useful property of the latter is the existence of identities, obtained by taking derivatives of the logarithm of the partition sum. It is shown that these identities still exist for models belonging to generalised exponential families, in which case they involve escort probability distributions. The percolation model serves as an example. A previously known identity is derived. It relates the average number of sites belonging to the finite cluster at the origin, the average number of perimeter sites, and the derivative of the order parameter.

cond-mat.stat-mech↗
arXiv2005

On the hydrino state of the relativistic hydrogen atom

The Klein-Gordon equation of the hydrogen atom has a low-lying eigenstate, called hydrino state, with square integrable wavefunction. The corresponding spinor solution of Dirac's equation is not square integrable. For this reason the hydrino state has been rejected in the early days of quantum mechanics as being unphysical. Maybe it is time to change opinion.

physics.gen-ph↗
arXiv2004

Boltzmann entropy and the microcanonical ensemble

Boltzmann's entropy is slightly modified to make it suitable for discussing phase transitions in finite systems. As an example it is shown that the pendulum undergoes a second order phase transition when passing from a vibrational to a rotating state.

cond-mat.stat-mech↗