Remarks on entropy and irreversibility in non-hamiltonian systems

Entropy and irreversibility in non-hamiltonian systems are discussed from a fully covariant point of view. The discussion is based on the generalized Liouville equation, the definition of a volume element dV in state space, and a generalized hypothesis of equal a priori probabilities. Systems may be classified as reversible or irreversible according to whether it is or is not possible to define dV so that the flow is globally volume-preserving in state space.     

Nonanalyticities of entropy functions of finite and infinite systems

In contrast to the canonical ensemble where thermodynamic functions are smooth for all finite system sizes, the microcanonical entropy can show nonanalytic points also for finite systems. The relation between finite and infinite system nonanalyticities is illustrated by means of a simple classical spinlike model which is exactly solvable for both finite and infinite system sizes, showing a phase transition in the latter case. The microcanonical entropy is found to have exactly one nonanalytic point in the interior of its domain. For all finite system sizes, this point is located at the same fixed energy value epsilon(c)(finite), jumping discontinuously to a different value epsilon(c)(infinite) in the thermodynamic limit. Remarkably, epsilon(c)(finite) equals the average potential energy of the infinite system at the phase transition point. The result indicates that care is required when trying to infer infinite system properties from finite system nonanalyticities.     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Remarks on the entropy of 3-manifolds

We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are homeomorphic to S3. We discuss the problem of proving that all 3-dimensional simplicial spheres can be obtained by this construction and give an example of a simplicial 3-ball whose boundary triangles can be identified pairwise such that no triangle is identified with any of its neighbours and the resulting 3-dimensional simplicial complex is a simply connected 3-manifold.     

On the entropy of systems undergoing time evolution

The aim of the present paper is to discuss the behaviour of entropy of physical systems undergoing time evolution. We discuss the case of an infinite-dimensional separable Hilbert space $\text{H}$ where the entropy of a microstate ? is given by the formula s(?) = ?Tr(?ln?). Information about a physical system is given by the mean values Tr(?A_i) = m, i = 1, …, N, of N self-adjoint (not necessarily bounded) operators A_i.     

An infinite lie group of symmetry of one-dimensional gas flow,for a class of entropy distributions

We have shown in earlier works the existence of three previously unknown symmetries of the equations of one-dimensional gas dynamics, with arbitrary entropy distribution and arbitrary polytropic index γ. These symmetries are seen here to form a group whenever the equation of state is of the form P = ?³(a₀ + a₁M + a₂M²)^?2 where M = ∝?dr is the Lagrangian mass coordinate.Introducing the remaining symmetry of space-translation enlarges the group into a Lie group of symmetry of infinite order, from which an infinite number of conservation laws can be deduced by application of Noether's theorem. The Lie group has a finite sub-algebra of order eight, which has SU3 structure; the list of associated conservation laws includes each of the six ones that are derivable from general physical principles, namely: the energy, momentum and the center-of-mass integrals, two integrals expressing scale invariance, and one associated with the virial theorem; the remaining two integrals of the octet are of a new type.Such a situation reminds us of the case of the Korteweg-de Vries equation in the soliton problem, where the symmetries and infinite number of conservation laws arise as a result of the possibility to linearize through the inverse-scattering method. Thus the question is raised of whether the inverse-scattering method also applies to gas-dynamical equations (with the above equation of state), or else whether another method of linearization may be found.     

On the linear response theory of infinite quantum systems

The perturbation of a C¹-dynamics α by a time-dependent unbounded 1-derivation of the form μ?(t)δ′ is considered. The existence of the perturbed dynamics and that of the linear responce oftthe C¹-dynamical system ($\text{A}$, $\text{R}$, α) is shown. The existence of the corresponding generalized susceptibility and some convergence problems are discussed.     

Modelling one-dimensional driven diffusive systems by the Zero-Range Process

In order to characterize networks in the scale-free network class we study the frequency of cycles of length h that indicate the ordering of network structure and the multiplicity of paths connecting two nodes. In particular we focus on the scaling of the number of cycles with the system size in off-equilibrium scale-free networks. We observe that each off-equilibrium network model is characterized by a particular scaling in general not equal to the scaling found in equilibrium scale-free networks. We claim that this anomalous scaling can occur in real systems and we report the case of the Internet at the Autonomous System Level.Received: 15 January 2004, Published online: 14 May 2004PACS: 89.75.-k Complex systems - 89.75.Hc Networks and genealogical trees     

Remarks to the topological characterization of cellular systems

In this note we discuss and solve an open problem (a conjecture) posed in a paper “On the combinatorial characterization of quasicrystals” published earlier in the Journal of Geometry and Physics. The conjecture proved is valid for the majority of space filling cellular systems (polycrystals, nanotubes, and fullerenes).     

On the nature of the phase transition in the KDP systems

Using the Landau theory of phase transitions it has been shown that for a second order phase transition Ω/k_BT_c ? 0.01. and its isomorphs 4Ω/J¹ ? 1 and for a first order transition Ω/k_BT_c ? 0.01.     

Irreversibilities in a triple diffusive flow in various porous cavities

Entropy generation minimization approach is a very good method allowing to analyze the engineering systems to exclude technical failure. The present study deals with computational analysis of triple diffusive flow, energy transference and entropy production in different porous cavities from square to triangular through trapezoidal shape. The formulated boundary-value problem has been worked out using the finite element technique and non-primitive variables. The developed computational code has been verified using numerical results of other researchers. Analysis of entropy production due to energy and mass transport, motion friction, and porous material has been performed for different chamber's shapes. Entropy generation analysis in chambers of various geometries under the triple-diffusive flow is a novelty of the present research, where different entropy production mechanisms have been scrutinized for one complex problem. It has been ascertained that average total entropy generation strength raises with buoyancy ratios, Lewis and Rayleigh numbers, but it has the minimum value for the square chamber in comparison with triangular and trapezoidal shapes. Moreover, obtained results characterize a neglecting influence of motion friction on the total entropy generation.     

On the number of infinite geodesics and ground states in disordered systems

We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to +. As corollaries we show that in any dimensiond2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or +. Proofs employ simple large-deviation estimates and ergodic arguments.     

Velocity of a perturbation in infinite lattice systems

The problem of defining and estimating the velocity of disturbances in a crystal is investigated. Some results are given for plane rotors and anharmonic systems.Research partially supported by CNR.     

On non-equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case

It is shown that for a set of full measure with respect to any translation invariant probability distribution on the space of initial configurarations of classical particle systems on ^d with interaction given by a smooth superstable potential of finite range there is a solution to the Newtonian equations of motion, provided that the specific energy and the particle density of the initial configuration exist a.s.     

Semiclassical approach to orbital magnetism of interacting diffusive quantum systems

We study interaction effects on the orbital magnetism of diffusive mesoscopic quantum systems. By combining many-body perturbation theory with semiclassical techniques, we show that the interaction contribution to the ensemble-averaged quantum thermodynamic potential can be reduced to an essentially classical operator. We compute the magnetic response of disordered rings and dots for diffusive classical dynamics. Our semiclassical approach reproduces the results of previous diagrammatic quantum calculations.     

On singularities in the disordered phase of a driven diffusive system

Using renormalization group techniques, we investigate the large distance behavior of a driven, interacting lattice gas in the disordered phase. Unlike the equilibrium Ising model, its behavior, ford>2, is controlled by aline of fixed points, each of which is interpreted as a dynamical system violating the fluctuation-dissipation theorem (FDT). As a consequence, correlation functions at large distances typically decay according to a power law instead of an exponential. Ford2, the renormalization group flows towards an FDT-satisfying fixed point, which corresponds to the high-temperature, strong-drive limit. In the steady state of such a model (a driven, free lattice gas), correlations are known to be exactly zero. Nevertheless, our correlations are still dominated by power laws, since the FDT-breaking operators aredangerously irrelevant (marginal ind=2). Thus, for anyd, the long wavelength properties cannot be obtained by taking either the non-interacting or theT limit, unlike for the equilibrium Ising model.