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Jan Naudts

Publications and source records attributed to Jan Naudts.

At least 73 records · Page 4Linked to original sources
arXiv2004

Escort density operators and generalized quantum information measures

Parametrized families of density operators are studied. A generalization of the lower bound of Cramer and Rao is formulated. It involves escort density operators. The notion of phi-exponential family is introduced. This family, together with its escort, optimizes the generalized lower bound. It also satisfies a maximum entropy principle and exhibits a thermodynamic structure in which entropy and free energy are related by Legendre transform.

cond-mat.stat-mech↗
arXiv2004

On the maximum entropy principle in non-extensive thermostatistics

It is possible to derive the maximum entropy principle from thermodynamic stability requirements. Using as a starting point the equilibrium probability distribution, currently used in non-extensive thermostatistics, it turns out that the relevant entropy function is Renyi's alpha-entropy, and not Tsallis' entropy.

cond-mat.stat-mech↗
arXiv2004

Continuity of a class of entropies and relative entropies

The present paper studies continuity of generalized entropy functions and relative entropies defined using the notion of a deformed logarithmic function. In particular, two distinct definitions of relative entropy are discussed. As an application, all considered entropies are shown to satisfy Lesche's stability condition. The entropies of Tsallis' nonextensive thermostatistics are taken as examples.

math-ph↗
arXiv2003

Generalized thermostatistics based on deformed exponential and logarithmic functions

The equipartition theorem states that inverse temperature equals the log-derivative of the density of states. This relation can be generalized by introducing a proportionality factor involving an increasing positive function phi(x). It is shown that this assumption leads to an equilibrium distribution of the Boltzmann-Gibbs form with the exponential function replaced by a deformed exponential function. In this way one obtains a formalism of generalized thermostatistics introduced previously by the author. It is shown that Tsallis' thermostatistics, with a slight modification, is the most obvious example of this formalism and corresponds with the choice phi(x)=x^q.

cond-mat.stat-mech↗
arXiv2003

Generalized thermostatistics and mean-field theory

The present paper studies a large class of temperature dependent probability distributions and shows that entropy and energy can be defined in such a way that these probability distributions are the equilibrium states of a generalized thermostatistics. This generalized thermostatistics is obtained from the standard formalism by deformation of exponential and logarithmic functions. Since this procedure is non-unique, specific choices are motivated by showing that the resulting theory is well-behaved. In particular, the equilibrium state of any system with a finite number of degrees of freedom is, automatically, thermodynamically stable and satisfies the variational principle. The equilibrium probability distribution of open systems deviates generically from the Boltzmann-Gibbs distribution. If the interaction with the environment is not too strong then one can expect that a slight deformation of the exponential function, appearing in the Boltzmann-Gibbs distribution, can reproduce the observed temperature dependence. An example of a system, where this statement holds, is a single spin of the Ising chain. The connection between the present formalism and Tsallis' thermostatistics is discussed. In particular, the present generalization sheds some light onto the historical development of the latter formalism.

cond-mat.stat-mech↗
arXiv2003

Non-unique way to generalize the Boltzmann-Gibbs distribution

Alternative definitions are given of basic concepts of generalized thermostatistics. In particular, generalizations of Shannon's entropy, of the Boltzmann-Gibbs distribution, and of relative entropy are considered. Particular choices made in Tsallis' nonextensive thermostatistics are questioned.

cond-mat.stat-mech↗
arXiv2002

Photon fields in a fluctuating spacetime

We present a model of interacting quantum fields, formulated in a non-perturbative manner. One of the fields is treated semi-classically, the other is the photon field. The model has an interpretation of an electromagnetic field in a fluctuating spacetime. The model is equivalent with the quantization of electromagnetism proposed recently by Czachor. Interesting features are that standard photon theory is recovered as a limiting case, and that localized field operators for the electromagnetic field exist as unbounded operators in Hilbert space.

hep-th↗
arXiv2002

Deformed exponentials and logarithms in generalized thermostatistics

Criteria are given that kappa-deformed logarithmic and exponential functions should satisfy. With a pair of such functions one can associate another function, called the deduced logarithmic function. It is shown that generalized thermostatistics can be formulated in terms of kappa-deformed exponential functions together with the associated deduced logarithmic functions.

cond-mat.stat-mech↗
arXiv2002

Covariance approach to the free photon field

We introduce photon theory following the same principles as for introduction of the quantum theory of a single particle, using a C*-algebraic approach based on covariance systems. The basic symmetries are additivity of the fields and additivity of test functions. We write down in explicit form a state of this covariance system. It turns out to reproduce the traditional Fock representation of the free photon field, with a Lorentz invariant vacuum. Properties of smeared-out photons are discussed.

quant-ph↗
arXiv2001

Generalized thermostatistics and Kolmogorov-Nagumo averages

We introduce a generalized thermostatistics based on Kolmogorov-Nagumo averages and appropriately selected information measures. The formalism includes Tsallis non-extensive thermostatistics, but also extensive thermostatistics based on Renyi entropy. The Curie-Weiss model is discussed as an example.

cond-mat.stat-mech↗
arXiv2001

Thermodynamic origin of universal fluctuations and two-power laws

We discuss universality of response functions in systems with excited degrees of freedom. We propose a unification of two existing phenomenologies, two-power law decay and deviation from power law due to non-extensivity. A universal curve is derived by maximizing entropy with a non-linear constraint. The same formalism can explain the universal fluctuation curves which have been discovered recently by Bramwell, Holdsworth, and Pinton.

cond-mat.stat-mech↗
arXiv2000

Operator formalism of quantum mechanics

This is the first chapter of a new and unconventional textbook on quantum mechanics and quantum field theory. The chapter introduces standard quantum mechanics by means of a symmetry principle, without reference to classical mechanics. The mathematical foundation of this approach comes from a recent paper of Naudts and Kuna on covariance systems. The standard representation of quantum mechanics is derived. Next, spin and mass of a quantum particle are explained as labels of projective representations of the Galilei group.

math-ph↗
arXiv2000

Model of a quantum particle in spacetime

Doplicher, Fredenhagen, and Roberts (1994, 1995) proposed a simple model of a particle in quantum spacetime. We give a new formulation of the model and propose some small changes and additions which improve the physical interpretation. In particular, we show that the internal degrees of freedom e and m of the particle represent external forces acting on the particle. To obtain this result we follow a constructive approach. The model is formulated as a covariance system. It has projective representations in which not only the spacetime coordinates but also the conjugated momenta are two-by-two noncommuting. These momenta are of the form P_mu-(b/c)A_mu, where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential A_mu coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Q_mu-(al^2/hbar c) Omega_mu where a is a generalized charge, l a fundamental length, and with vector potentials Omega_mu which are in some sense dual w.r.t. the A_mu.

hep-th↗
arXiv2000

Solving inverse problems by combination of maximum entropy and montecarlo simulation

The montecarlo method, which is quite commonly used to solve maximum entropy problems in statistical physics, can actually be used to solve inverse problems in a much wider context. The probability distribution which maximizes entropy can be calculated analytically by introducing Lagrange parameters. The problem of fixing these lagrangean parameters is circumvented by introduction of a microcanonical ensemble which describes a system together with its heath bath. Some further simplifying assumptions make it feasible to do montecarlo sampling of the probability distribution. The method is applied to the example of determining the distribution of the density of the earth from three data. Advantages of the method are guaranteed convergence and a clear information-theoretic foundation.

cond-mat.stat-mech↗
arXiv2000

Covariance systems

We introduce new definitions of states and of representations of covariance systems. The GNS-construction is generalized to this context. It associates a representation with each state of the covariance system. Next, states are extended to states of an appropriate covariance algebra. Two applications are given. We describe a nonrelativistic quantum particle, and we give a simple description of the quantum spacetime model introduced by Doplicher et al.

math-ph↗