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A. G. Bashkirov

Publications and source records attributed to A. G. Bashkirov.

7 recordsLinked to original sources
arXiv2006

Renyi Thermostatistics and Self-Organization

Taking into account extremum of a Helmholtz free energy in the equilibrium state of a thermodynamic system the Renyi entropy is derived from the Boltzmann entropy by the same way as the Helmholtz free energy from the Hamiltonian. The application of maximum entropy principle to the Renyi entropy gives rise to the Renyi distribution. The $q$-dependent Renyi thermodynamic entropy is defined as the Renyi entropy for Renyi distribution. A temperature and free energy are got for a Renyi thermostatistics. Transfer from the Gibbs to Renyi thermostatistics is found to be a phase transition at zero value of an order parameter $η=1-q$. It is shown that at least for a particular case of the power-law Hamiltonian $H=C\sum_i x_i^κ$ this entropy increases with $η$. Therefore in the new entropic phase at $η>0$ the system tends to develop into the most ordered state at $η=η_{max}=κ/(1+κ)$. The Renyi distribution at $η_{max}$ becomes a pure power-law distribution.

cond-mat.stat-mech↗
arXiv2005

The Renyi entropy as a "free entropy" for complex systems

The Boltzmann entropy $S^{(B)}$ is true in the case of equal probability of all microstates of a system. In the opposite case it should be averaged over all microstates that gives rise to the Boltzmann--Shannon entropy (BSE). Maximum entropy principle (MEP) for the BSE leads to the Gibbs canonical distribution that is incompatible with power--low distributions typical for complex system. This brings up the question: Does the maximum of BSE correspond to an equilibrium (or steady) state of the complex system? Indeed, the equilibrium state of a thermodynamic system which exchange heat with a thermostat corresponds to maximum of Helmholtz free energy rather than to maximum of average energy, that is internal energy $U$. Following derivation of Helmholtz free energy the Renyi entropy is derived as a cumulant average of the Boltzmann entropy for systems which exchange an entropy with the thermostat. The application of MEP to the Renyi entropy gives rise to the Renyi distribution for an isolated system. It is investigated for a particular case of a power--law Hamiltonian. Both Lagrange parameters, $α$ and $β$ can be eliminated. It is found that $β$ does not depend on a Renyi parameter $q$ and can be expressed in terms of an exponent $κ$ of the power--law Hamiltonian and $U$. The Renyi entropy for the resulting Renyi distribution reaches its maximal value at $q=1/(1+κ)$ that can be considered as the most probable value of $q$ when we have no additional information on behavior of the stochastic process. The Renyi distribution for such $q$ becomes a power--law distribution with the exponent $-(κ+1)$. Such a picture corresponds to some observed phenomena in complex systems.

cond-mat.stat-mech↗
arXiv2004

Maximum Renyi entropy principle for systems with power--law Hamiltonian

The Renyi distribution ensuring the maximum of a Renyi entropy is investigated for a particular case of a power--law Hamiltonian. Both Lagrange parameters, $α$ and $β$ can be excluded. It is found that $β$ does not depend on a Renyi parameter $q$ and can be expressed in terms of an exponent $κ$ of the power--law Hamiltonian and an average energy $U$. The Renyi entropy for the resulted Renyi distribution reaches its maximal value at $q=1/(1+κ)$ that can be considered as the most probable value of $q$ when we have no additional information on behaviour of the stochastic process. The Renyi distribution for such $q$ becomes a power--law distribution with the exponent $-(κ+1)$. When $q=1/(1+κ)+ε$ ($0<ε\ll 1$) there appears a horizontal "head" part of the Renyi distribution that precedes the power--law part. Such a picture corresponds to observables.

cond-mat.stat-mech↗
arXiv2003

Maximum entropy principle for Renyi's and Tsallis' entropies

The equilibrium distributions of probabilities providing maximality of Renyi and Tsallis entropies are rederived. New S-forms of them are found which are normalised with corresponding entropies in contrast to the usual Z-forms normalised with partition functions.

cond-mat.stat-mech↗
arXiv2001

Dynamical Screening of Gravitational Interaction and Planetary Motions in Modified Solar Potential

A density disturbance in a system of gravitating mass, induced by a moving selected body gives rise to a dynamical screening of Newtonian potential of this body. When applied to the solar planetary system it means that as a result of the motion of the Sun in the Galaxy its effective force potential appears more weak than the Newtonian potential. The relevant modifications of main relations of the solar dynamics are considered here and it is found in particular that the reestimated period of the Earth revolution around the Sun rises in 1 second per year and semimajor axis of the Earth orbit increases on 4 kilometers. Similar relations are obtained for other planets too. It may be supposed that the inclusion of these effects can help to explain the observable anomalous acceleration of spacecrafts Pioneer 10 and 11.

astro-ph↗
arXiv2001

Thermodynamics of Coherent States and Black Hole Entropy

Entropy and temperature of a system in a coherent state are naturally defined on a base of a density matrix of the system. As an example, entropy and temperature are evaluated for coherent states of a harmonic oscillator and quantum field described by the Klein--Gordon--Fock equation with a source term. It is shown, in particular, that the temperature of the coherent oscillator in a ground state coincides with the effective temperature of a harmonic oscillator being in contact with a heat bath (Bloch formula) when the bath temperature tends to zero. The Bekenstein--Hawking entropy of a black hole can also be interpreted as an entropy of coherent states of a physical vacuum in the vicinity of a horizon surface.

quant-ph↗