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Papers by Andrzej Ostruszka

Research paper thumbnail of Quantization of classical maps with tunable Ruelle-Pollicott resonances

Physical review. E, Statistical, nonlinear, and soft matter physics, 2003

We investigate the correspondence between the decay of correlation in classical systems, governed... more We investigate the correspondence between the decay of correlation in classical systems, governed by Ruelle-Pollicott resonances, and the properties of the corresponding quantum systems. For this purpose we construct classical dynamics with controllable resonances together with their quantum counterparts. As an application of such tunable resonances we reveal the role of Ruelle-Pollicott resonances for the localization properties of quantum energy eigenstates.

Research paper thumbnail of Renyi Entropies in Particle Cascades

Acta Physica Polonica B, 2003

Renyi entropies for particle distributions following from the general cascade models are discusse... more Renyi entropies for particle distributions following from the general cascade models are discussed. The p-model and the β distribution introduced in earlier studies of cascades are discussed in some detail. Some phenomenological consequences are pointed out.

Research paper thumbnail of Quantum baker map on the sphere

Nonlinearity, 1998

We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The... more We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a family of the corresponding quantum maps. In the simplest case of the model the system does not possess a time reversal symmetry and the quantum map is represented by real, orthogonal matrices of even dimension. The semiclassical ensemble of quantum maps, obtained by averaging over a range of matrix sizes, displays statistical properties characteristic of circular unitary ensemble. Time evolution of such systems may be studied with the help of the SU (2) coherent states and the generalized Husimi distribution.

Research paper thumbnail of Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitativ... more Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the Kolmogorov-Sinai ͑KS͒ entropy diverges if the diameter of the partition tends to zero, we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is finite and non-negative and we call it the dynamical entropy of the noisy system. In the weak noise limit this quantity is conjectured to tend to the KS entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel for which the Frobenius-Perron operator can be represented by a finite matrix.

Research paper thumbnail of Spectrum of the Frobenius–Perron operator for systems with stochastic perturbation

Physics Letters A, 2001

We investigate dynamical systems with stochastic perturbation and study to what extend analytical... more We investigate dynamical systems with stochastic perturbation and study to what extend analytical properties of the noise present influence the spectrum of the associated Frobenius-Perron operator. We suggest to distinguish a "physical" part of the spectrum of the deterministic system, as this robust with respect to the perturbation. For exemplary system studied such eigenvalues of the FP-operator are located outside the essential spectrum and have direct physical meaning: they determine the rate of the exponential decay of correlations in the system. 

Research paper thumbnail of Basis Markov Partitions and Transition Matrices for Stochastic Systems

Siam Journal on Applied Dynamical Systems, 2008

We analyze dynamical systems subjected to an additive noise and their deterministic limit. In thi... more We analyze dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a stochastic system has something like a Markov partition for deterministic systems. For a chosen class of the noise profiles the Frobenius-Perron operator associated to the noisy system is exactly represented by a stochastic transition matrix of a finite size K. This feature allows us to introduce for these stochastic systems a basis--Markov partition, defined herein, irrespectively of whether the deterministic system possesses a Markov partition or not. We show that in the deterministic limit, corresponding to K --> infinity, the sequence of invariant measures of the noisy systems tends, in the weak sense, to the invariant measure of the deterministic system. Thus by introducing a small additive noise one may approximate transition matrices and invariant measures of deterministic dynamical systems.

Research paper thumbnail of Basis Markov Partitions and Transition Matrices for Stochastic Systems

We analyze dynamical systems subjected to an additive noise and their deterministic limit. In thi... more We analyze dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a stochastic system has something like a Markov partition for deterministic systems. For a chosen class of the noise profiles the Frobenius-Perron operator associated to the noisy system is exactly represented by a stochastic transition matrix of a finite size K. This feature allows us to introduce for these stochastic systems a basis--Markov partition, defined herein, irrespectively of whether the deterministic system possesses a Markov partition or not. We show that in the deterministic limit, corresponding to K --> infinity, the sequence of invariant measures of the noisy systems tends, in the weak sense, to the invariant measure of the deterministic system. Thus by introducing a small additive noise one may approximate transition matrices and invariant measures of deterministic dynamical systems.

Research paper thumbnail of Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitativ... more Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the Kolmogorov-Sinai ͑KS͒ entropy diverges if the diameter of the partition tends to zero, we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is finite and non-negative and we call it the dynamical entropy of the noisy system. In the weak noise limit this quantity is conjectured to tend to the KS entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel for which the Frobenius-Perron operator can be represented by a finite matrix.

Research paper thumbnail of Quantization of classical maps with tunable Ruelle-Pollicott resonances

Physical Review E, 2003

We investigate the correspondence between the decay of correlation in classical systems, governed... more We investigate the correspondence between the decay of correlation in classical systems, governed by Ruelle-Pollicott resonances, and the properties of the corresponding quantum systems. For this purpose we construct classical dynamics with controllable resonances together with their quantum counterparts. As an application of such tailormade resonances we reveal the role of Ruelle-Pollicott resonances for the localization properties of quantum energy eigenstates.

Research paper thumbnail of Dynamical entropy for systems with stochastic perturbation

Physical Review E, 2000

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitativ... more Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the KS-entropy diverges we analyse the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is non negative and in the weak noise limit is conjectured to tend to the KS-entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel, for which the Frobenius-Perron operator can be represented by a finite matrix.

Research paper thumbnail of Quantization of classical maps with tunable Ruelle-Pollicott resonances

Physical review. E, Statistical, nonlinear, and soft matter physics, 2003

We investigate the correspondence between the decay of correlation in classical systems, governed... more We investigate the correspondence between the decay of correlation in classical systems, governed by Ruelle-Pollicott resonances, and the properties of the corresponding quantum systems. For this purpose we construct classical dynamics with controllable resonances together with their quantum counterparts. As an application of such tunable resonances we reveal the role of Ruelle-Pollicott resonances for the localization properties of quantum energy eigenstates.

Research paper thumbnail of Renyi Entropies in Particle Cascades

Acta Physica Polonica B, 2003

Renyi entropies for particle distributions following from the general cascade models are discusse... more Renyi entropies for particle distributions following from the general cascade models are discussed. The p-model and the β distribution introduced in earlier studies of cascades are discussed in some detail. Some phenomenological consequences are pointed out.

Research paper thumbnail of Quantum baker map on the sphere

Nonlinearity, 1998

We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The... more We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a family of the corresponding quantum maps. In the simplest case of the model the system does not possess a time reversal symmetry and the quantum map is represented by real, orthogonal matrices of even dimension. The semiclassical ensemble of quantum maps, obtained by averaging over a range of matrix sizes, displays statistical properties characteristic of circular unitary ensemble. Time evolution of such systems may be studied with the help of the SU (2) coherent states and the generalized Husimi distribution.

Research paper thumbnail of Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitativ... more Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the Kolmogorov-Sinai ͑KS͒ entropy diverges if the diameter of the partition tends to zero, we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is finite and non-negative and we call it the dynamical entropy of the noisy system. In the weak noise limit this quantity is conjectured to tend to the KS entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel for which the Frobenius-Perron operator can be represented by a finite matrix.

Research paper thumbnail of Spectrum of the Frobenius–Perron operator for systems with stochastic perturbation

Physics Letters A, 2001

We investigate dynamical systems with stochastic perturbation and study to what extend analytical... more We investigate dynamical systems with stochastic perturbation and study to what extend analytical properties of the noise present influence the spectrum of the associated Frobenius-Perron operator. We suggest to distinguish a "physical" part of the spectrum of the deterministic system, as this robust with respect to the perturbation. For exemplary system studied such eigenvalues of the FP-operator are located outside the essential spectrum and have direct physical meaning: they determine the rate of the exponential decay of correlations in the system. 

Research paper thumbnail of Basis Markov Partitions and Transition Matrices for Stochastic Systems

Siam Journal on Applied Dynamical Systems, 2008

We analyze dynamical systems subjected to an additive noise and their deterministic limit. In thi... more We analyze dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a stochastic system has something like a Markov partition for deterministic systems. For a chosen class of the noise profiles the Frobenius-Perron operator associated to the noisy system is exactly represented by a stochastic transition matrix of a finite size K. This feature allows us to introduce for these stochastic systems a basis--Markov partition, defined herein, irrespectively of whether the deterministic system possesses a Markov partition or not. We show that in the deterministic limit, corresponding to K --> infinity, the sequence of invariant measures of the noisy systems tends, in the weak sense, to the invariant measure of the deterministic system. Thus by introducing a small additive noise one may approximate transition matrices and invariant measures of deterministic dynamical systems.

Research paper thumbnail of Basis Markov Partitions and Transition Matrices for Stochastic Systems

We analyze dynamical systems subjected to an additive noise and their deterministic limit. In thi... more We analyze dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a stochastic system has something like a Markov partition for deterministic systems. For a chosen class of the noise profiles the Frobenius-Perron operator associated to the noisy system is exactly represented by a stochastic transition matrix of a finite size K. This feature allows us to introduce for these stochastic systems a basis--Markov partition, defined herein, irrespectively of whether the deterministic system possesses a Markov partition or not. We show that in the deterministic limit, corresponding to K --> infinity, the sequence of invariant measures of the noisy systems tends, in the weak sense, to the invariant measure of the deterministic system. Thus by introducing a small additive noise one may approximate transition matrices and invariant measures of deterministic dynamical systems.

Research paper thumbnail of Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitativ... more Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the Kolmogorov-Sinai ͑KS͒ entropy diverges if the diameter of the partition tends to zero, we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is finite and non-negative and we call it the dynamical entropy of the noisy system. In the weak noise limit this quantity is conjectured to tend to the KS entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel for which the Frobenius-Perron operator can be represented by a finite matrix.

Research paper thumbnail of Quantization of classical maps with tunable Ruelle-Pollicott resonances

Physical Review E, 2003

We investigate the correspondence between the decay of correlation in classical systems, governed... more We investigate the correspondence between the decay of correlation in classical systems, governed by Ruelle-Pollicott resonances, and the properties of the corresponding quantum systems. For this purpose we construct classical dynamics with controllable resonances together with their quantum counterparts. As an application of such tailormade resonances we reveal the role of Ruelle-Pollicott resonances for the localization properties of quantum energy eigenstates.

Research paper thumbnail of Dynamical entropy for systems with stochastic perturbation

Physical Review E, 2000

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitativ... more Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the KS-entropy diverges we analyse the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is non negative and in the weak noise limit is conjectured to tend to the KS-entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel, for which the Frobenius-Perron operator can be represented by a finite matrix.