arXiv subjects

S. M. Apenko

Publications and source records attributed to S. M. Apenko.

8 recordsLinked to original sources

arXiv2014

Clausius inequality and H-theorems for some models of random wealth exchange

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution equation that describes random wealth exchanges. In such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy. It appears useful to reformulate the problem and represent the dynamics as a combination of two processes. The first is a linear transformation of a two-particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control. This representation leads to a Clausius-type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type. Only in some special cases when equilibrium distribution is exactly a gamma distribution, this inequality results in the H-theorem with monotonically growing `entropy' functional which differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule the evolution has some features of relaxation to a non-equilibrium steady state and it is still unclear if any general H-theorem could exist.

cond-mat.stat-mech↗

arXiv2013

In Search of H-theorem for Ulam's Redistribution Problem

We discuss the possibility of deriving an H-theorem for the nonlinear discrete time evolution known as Ulam's redistribution of energy problem. In this model particles are paired at random and then their total energy is redistributed between them according to some probability law. It appears possible to obtain the proper H-function which always increases during the relaxation only for a special set of redistribution laws, given by symmetric beta distributions. This H-function differs from the usual entropy by an additional term that vanishes only for the uniform redistribution law. But for arbitrary redistribution the evolution has some features of relaxation to a non-equilibrium steady state and the H-function is still unknown.

cond-mat.stat-mech↗

arXiv2013

Monotonic entropy growth for a nonlinear model of random exchanges

We present a proof of the monotonic entropy growth for a nonlinear discrete-time model of a random market. This model, based on binary collisions, also may be viewed as a particular case of Ulam's redistribution of energy problem. We represent each step of this dynamics as a combination of two processes. The first one is a linear energy-conserving evolution of the two-particle distribution, for which the entropy growth can be easily verified. The original nonlinear process is actually a result of a specific `coarse-graining' of this linear evolution, when after the collision one variable is integrated away. This coarse-graining is of the same type as the real space renormalization group transformation and leads to an additional entropy growth. The combination of these two factors produces the required result which is obtained only by means of information theory inequalities.

cond-mat.stat-mech↗

arXiv2011

Information theory and renormalization group flows

We present a possible approach to the study of the renormalization group (RG) flow based entirely on the information theory. The average information loss under a single step of Wilsonian RG transformation is evaluated as a conditional entropy of the fast variables, which are integrated out, when the slow ones are held fixed. Its positivity results in the monotonic decrease of the informational entropy under renormalization. This, however, does not necessarily imply the irreversibility of the RG flow, because the entropy explicitly depends on the total number of degrees of freedom, which is reduced. Only some size-independent additive part of the entropy could possibly provide the required Lyapunov function. We also introduce a mutual information of fast and slow variables as probably a more adequate quantity to represent the changes in the system under renormalization and evaluate it for some simple systems. It is shown that for certain real space decimation transformations the positivity of the mutual information directly leads to the monotonic growth of the entropy per lattice site along the RG flow and hence to its irreversibility.

cond-mat.stat-mech↗

arXiv2007

Renormalization of the vacuum angle in quantum mechanics, Berry phase and continuous measurements

The vacuum angle $θ$ renormalization is studied for a toy model of a quantum particle moving around a ring, threaded by a magnetic flux $θ$. Different renormalization group (RG) procedures lead to the same generic RG flow diagram, similar to that of the quantum Hall effect. We argue that the renormalized value of the vacuum angle may be observed if the particle's position is measured with finite accuracy or coupled to additional slow variable, which can be viewed as a coordinate of a second (heavy) particle on the ring. In this case the renormalized $θ$ appears as a magnetic flux this heavy particle sees, or the Berry phase, associated with its slow rotation.

hep-th↗

arXiv2006

Renormalization of the vacuum angle for a particle on a ring

We analyze the vacuum (topological) angle $θ$ renormalization for the quantum mechanical model of a particle moving around a ring, where $θ$ is the magnetic flux through the ring. We construct a renormalization group (RG) transformation for this model and derive exact RG equations which lead to the flow diagram similar to that of the Quantum Hall effect. Renormalization of $θ$ is seen to follow from the loss of information about the initial topological charge in the course of the RG procedure.

cond-mat.mes-hall↗

arXiv2000

Superfluid transition temperature from the Lindemann-like criterion

We further analyze the recently proposed criterion, according to which a superfluid transition occurs whenever the r.m.s. displacement of a particle from its initial position in imaginary time reaches a certain fraction of the interparticle distance. The critical temperature $T_λ$ is expressed in terms of the single particle mobility. Within the Drude approximation for the mobility $T_λ$ is determined by the friction coefficient (or by the viscosity for dense liquids). To check the validity of the criterion the resulting formula is applied to non-ideal Bose gas, to liquid helium and to $^3$He-$^4$He mixtures. A reasonable qualitative agreement with available experiments is obtained in all cases. In the case of the Bose gas the present approach results also in an upper bound on the peak value of $T_λ/T_λ^0$, where $T_λ^0$ is the critical temperature for the ideal gas.

cond-mat.stat-mech↗

arXiv1998

Weakly bound states in 2+εdimensions

We study the critical behaviour near the threshold where a first bound state appears at some value of coupling constant in an attractive short-range potential in $2+ε$ dimensions. We obtain general expression for the binding energy near the threshold and also demonstrate that the critical region is correctly described by an effective separable potential. The critical exponent of the radius of weakly bound state is shown to coincide with the correlation length exponent for the spin model in the large-N limit. In two dimensions, where the binding energy is exponentially small in coupling constant, we obtain a general analytic expression for the prefactor.

quant-ph↗