Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Quasiaverages and classification of equilibrium states of condensed media with spontaneously broken symmetry

Classification of equilibrium states of condensed media with spontaneously broken symmetry is carried out. Conditions of residual symmetry and spatial symmetry are formulated. The connection between these symmetry conditions and equilibrium states of various media with tensor order parameter is found out. Superfluid ³He, liquid crystals, quadrupolar magnetics are considered in detail. Possible homogeneous and heterogeneous states are found out. Discrete and continuous thermodynamic parameters, which define an equilibrium state, allowable form of order parameter, residual symmetry, and spatial symmetry generators are established. The text was submitted by the authors in English.     

Gleichgewichtsbedingungen in der Gittertheorie

The equilibrium conditions of an infinite crystal lattice result in symmetry relations of lattice sums. This was first shown byKun Huang who obtained these relations by comparing the linear theory of elasticity with the corresponding limit of lattice theory, a procedure which has been criticized by some authors. It is shown here how one can obtainKun Huang's results by using only trivial invariance properties of the potential energy. The same method is applied to get equilibrium relations for higher order (anharmonic) terms of lattice theory which are symmetry relations of tensors used in non linear elastic theory.     

A kinetic approach to some quasi-linear laws of macroeconomics

Some previous works have presented the data on wealth and income distributions in developed countries and have found that the great majority of population is described by an exponential distribution, which results in idea that the kinetic approach could be adequate to describe this empirical evidence. The aim of our paper is to extend this framework by developing a systematic kinetic approach of the socio-economic systems and to explain how linear laws, modelling correlations between macroeconomic variables, may arise in this context. Firstly we construct the Boltzmann kinetic equation for an idealised system composed by many individuals (workers, officers, business men, etc.), each of them getting a certain income and spending money for their needs. To each individual a certain time variable amount of money is associated - this meaning him/her phase space coordinate. In this way the exponential distribution of money in a closed economy is explicitly found. The extension of this result, including states near the equilibrium, give us the possibility to take into account the regular increase of the total amount of money, according to the modern economic theories. The Kubo-Green-Onsager linear response theory leads us to a set of linear equations between some macroeconomic variables. Finally, the validity of such laws is discussed in relation with the time reversal symmetry and is tested empirically using some macroeconomic time series. Received 25 February 2002 / Received in final form 11 July 2002 Published online 19 November 2002     

Rings of monopoles with discrete axial symmetry: explicit solution for N=3

It is shown that, in contrast to continuous axial symmetry, discrete axial symmetry admits separated SU(2) monopoles in static equilibrium. The Corrigan-Goddard conditions on the parameters are enormously simplified and for 3 equidistant monopoles are identically satisfied.     

Green's function approach to the quantum many-body theory of the solid state

TheGreen's function approach is used to develop a quantum many-body theory of the solid state which should work at low temperatures as well as in the neighbourhood of phase transition points. The theory is applicable also in those cases where the traditional expansion of the potential in powers of the atomic displacements is entirely inadequate (crystalline helium). The starting point of our approach is the concept of broken symmetry since the invariance of the equilibrium ensemble under the continuous group of infinitesimal translations is reduced in a crystalline solid to the invariance under finite translations through a lattice vector. A homogeneous integral equation is derived which has nontrivial solutions in the crystalline state. By this equation it is shown that the umklapp phonons are the symmetry restoring collective modes expected due to a general theorem ofGoldstone. The single particle excitations and the structure of the Dyson mass operator in the crystalline state are discussed. It is further shown that the homogeneous Bethe-Salpeter equation for the linear response to an external disturbance possesses symmetry breaking solutions which are connected to the lattice dynamics of the solid state. These collective excitations (phonons) are exhibited in RPA and tight-binding approximation for monoatomic cubic crystals with a Bravais lattice in order to demonstrate how the present theory reproduces well-known results.     

Classification in bifurcation theory and reaction-diffusion systems

The fundamental open problem in bifurcation theory is to determine when linearization and construction are valid. This problem and the problem of matching a critical exponent with a bifurcation point are solved through the application of selection rules. Selection rules are also central for a classification theory which is a natural extension of the above problems. Solutions are classified by four equivalence relations differing in their coarseness. Canonical function bases and bifurcation points are studied. The direction of increased vanishing of integrals involved in existence theorems correspond to various interesting similar order relations, for example to increased phase symmetry and decreased solution symmetry. The symmetries of physical and phase spaces are correlated and the closure of the solutions under the symmetry group is shown and analyzed. The common group theoretical basis for equilibrium and nonequilibrium transitions is emphasized throughout. Thus in both settings the same selection rules determine if the transition is continuous or discontinuous. A theory of symmetry breaking for nonequilibrium-bifurcation systems is described. After discussion of stability and jumps, the theory and the history of potentials in chemical systems far from equilibrium are reviewed from the mathematical, thermodynamic and catastrophe theory points of view. Implications to biological control, morphogenesis and pattern formation are briefly indicated. Throughout, reaction-diffusion is of central importance and it also serves as a carrier for the general ideas in bifurcation theory .     

Hydrodynamics and phases of flocks

We review the past decade’s theoretical and experimental studies of flocking: the collective, coherent motion of large numbers of self-propelled “particles” (usually, but not always, living organisms). Like equilibrium condensed matter systems, flocks exhibit distinct “phases” which can be classified by their symmetries. Indeed, the phases that have been theoretically studied to date each have exactly the same symmetry as some equilibrium phase (e.g., ferromagnets, liquid crystals). This analogy with equilibrium phases of matter continues in that all flocks in the same phase, regardless of their constituents, have the same “hydrodynamic”—that is, long-length scale and long-time behavior, just as, e.g., all equilibrium fluids are described by the Navier-Stokes equations. Flocks are nonetheless very different from equilibrium systems, due to the intrinsically nonequilibrium self-propulsion of the constituent “organisms.” This difference between flocks and equilibrium systems is most dramatically manifested in the ability of the simplest phase of a flock, in which all the organisms are, on average moving in the same direction (we call this a “ferromagnetic” flock; we also use the terms “vector-ordered” and “polar-ordered” for this situation) to exist even in two dimensions (i.e., creatures moving on a plane), in defiance of the well-known Mermin-Wagner theorem of equilibrium statistical mechanics, which states that a continuous symmetry (in this case, rotation invariance, or the ability of the flock to fly in any direction) can not be spontaneously broken in a two-dimensional system with only short-ranged interactions. The “nematic” phase of flocks, in which all the creatures move preferentially, or are simply oriented preferentially, along the same axis, but with equal probability of moving in either direction, also differs dramatically from its equilibrium counterpart (in this case, nematic liquid crystals). Specifically, it shows enormous number fluctuations, which actually grow with the number of organisms faster than the “law of large numbers” obeyed by virtually all other known systems. As for equilibrium systems, the hydrodynamic behavior of any phase of flocks is radically modified by additional conservation laws. One such law is conservation of momentum of the background fluid through which many flocks move, which gives rise to the “hydrodynamic backflow” induced by the motion of a large flock through a fluid. We review the theoretical work on the effect of such background hydrodynamics on three phases of flocks—the ferromagnetic and nematic phases described above, and the disordered phase in which there is no order in the motion of the organisms. The most surprising prediction in this case is that “ferromagnetic” motion is always unstable for low Reynolds-number suspensions. Experiments appear to have seen this instability, but a quantitative comparison is awaited. We conclude by suggesting further theoretical and experimental work to be done.     

Bunting identity and Mazur identity for non-linear elliptic systems including the black hole equilibrium problem

Recent work by G. Bunting and by P. O. Mazur has developed new techniques for proving uniqueness theorems for extensive classes of non-linear elliptic boundary value problems including that of the equilibrium state of an electromagnetically charged black hole. These methods are described and compared. It is shown that the rather general class of harmonic mappings that can be dealt with by the Bunting method (which needs no internal symmetry group) can be regarded as a generalisation of the particular (totally symetric) class of non-linear-models that can be dealt with by the Mazur method.     

Phase transitions for a continuous system of classical particles in a box

A continuous classical system involving an infinite number of distinguishable particles is analyzed along the same lines as its quantum analogue, considered in [1]. A commutativeC*-algebra is set up on the phase space of the system, and a representation-dependent definition of equilibrium involving the static KMS condition is given. For a special class of interactions the set of equilibrium states is realized as a convex Borel set whose extremal states are characterized by solutions to a system of integral equations. By analyzing these integral equations, we prove the absence of phase transitions for high temperature and construct a phase transition for low temperature. The construction also provides an example of a translation-invariant state whose decomposition at infinity yields states that are not translation-invariant. Thus we have an example in the classical situation of continuous symmetry breaking.This article is a part of the author's doctoral thesis, which was submitted to the mathematics department at Duke University     

Low-scale leptogenesis and the domain wall problem in models with discrete flavor symmetries

We propose a new mechanism for leptogenesis, which is naturally realized in models with a flavor symmetry based on the discrete group A₄$A_4$, where the symmetry breaking parameter also controls the Majorana masses for the heavy right-handed (RH) neutrinos. During the early universe, for T?TeV$T?\text{TeV}$, part of the symmetry is restored, due to finite temperature contributions, and the RH neutrinos remain massless and can be produced in thermal equilibrium even at temperatures well below the most conservative gravitino bounds. Below this temperature the phase transition occurs and they become massive, decaying out of equilibrium and producing the necessary lepton asymmetry. Unless the symmetry is broken explicitly by Planck-suppressed terms, the domain walls generated by the symmetry breaking survive till the quark–hadron phase transition, where they disappear due to a small energy splitting between the A₄$A_4$ vacua caused by the QCD anomaly.     

The Two-Particle Correlation Function for Systems with Long-Range Interactions

Pemantle and Steif provided a sharp threshold for the existence of a robust phase transition (RPT) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer matrix whose kernel describes the nearest neighbor interaction along the edges of the tree. Here a RPT is said to occur if an arbitrarily weak coupling with symmetry-breaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should occur at the same threshold in the continuous rotator model. We consider the class of four- and five-state rotation-invariant spin models with reflection symmetry on general trees which contains the Potts model and the clock model with scalarproduct-interaction as limiting cases. The clock model can be viewed as a particular discretization which is obtained from the classical rotator model with state space \(S^1\). We analyze the transition between \(\hbox {PT}=\hbox {RPT}\) and \(\hbox {PT}\ne \hbox {RPT}\), in terms of the eigenvalues of the transfer matrix of the model at the critical threshold value for the existence of RPT. The transition between the two regimes depends sensitively on the third largest eigenvalue.     

Ultrametricity between states at different temperatures in spin-glasses

We prove the existence of correlations between the equilibrium states at different temperatures of the multi-p-spin spherical spin-glass models with continuous replica symmetry breaking: there is no chaos in temperature in these models. Furthermore, the overlaps satisfy ultrametric relations. As a consequence the Parisi tree is essentially the same at all temperatures with lower branches developing when lowering the temperature. We conjecture that the reference free energies of the clusters are also fixed at all temperatures as in the generalized random-energy model. Received 18 March 2002 / Received in final form 14 June 2002 Published online 1st October 2002 RID="a" ID="a"e-mail: tommaso.rizzo@inwind.it     

Spin waves,vortices, and the structure of equilibrium states in the classicalXY model

We prove that, for spin systems with a continuous symmetry group on lattices of arbitrary dimension, the surface tension vanishes at all temperatures. For the classicalXY model in zero magnetic field, this result is shown to imply absence of interfaces in the thermodynamic limit, at arbitrary temperature. We show that, at values of the temperature at which the free energy of that model is continuously differentiable, i.e. at all except possibly countably many temperatures, there iseither aunique translation-invariant equilibrium state, or all such states are labelled by the elements of the symmetry group, SO(2). Moreover, there areno non-translation-invariant, but periodic equilibrium states. We also reconsider the representation of theXY model as a gas of spin waves and vortices and discuss the possibility that, in four or more dimensions, translation invariance may be broken by imposing boundary conditions which force an (open) vortex sheet through the system. Among our main tools are new correlation inequalities.     

Universal symmetry property of point defects in crystals

Point defects (vacancies, solute atoms, and disorder) are ubiquitous in crystalline solids. With in situ transmission electron microscopy we find clear evidence for the existence of a universal symmetry property of point defects; i.e., the symmetry of short-range order of point defects follows the crystal symmetry when in equilibrium. We further show that this symmetry-conforming property can lead to various interesting effects including "aging-induced microstructure memory" and the associated "time-dependent two-way shape memory."     

Statistical mechanics of money: how saving propensity affects its distribution

We consider a simple model of a closed economic system where the total money is conserved and the number of economic agents is fixed. Analogous to statistical systems in equilibrium, money and the average money per economic agent are equivalent to energy and temperature, respectively. We investigate the effect of the saving propensity of the agents on the stationary or equilibrium probability distribution of money. When the agents do not save, the equilibrium money distribution becomes the usual Gibb's distribution, characteristic of non-interacting agents. However with saving, even for individual self-interest, the dynamics becomes cooperative and the resulting asymmetric Gaussian-like stationary distribution acquires global ordering properties. Intriguing singularities are observed in the stationary money distribution in the market, as functions of the marginal saving propensity of the agents. Received 2 May 2000     

Real-time Bose-Einstein condensation in a finite volume with a discrete spectrum

We show that Bose condensation in real time occurs in a finite system not only as an accumulation of the bosons in the ground state below a critical temperature, but also as a rapid enhancement of an arbitrary small symmetry breaking, followed by a very slow decay of the symmetry breaking order parameter from the almost ideal value to the vanishing equilibrium value. We show this analytically on an exactly soluble model and numerically on a model of noninteracting bosons in an oscillator potential.     

On the absence of spontaneous breakdown of continuous symmetry for equilibrium states in two dimensions

Using the Bogoliubov inequality, we extend previously known results concerning the absence of continuous symmetry breakdown for equilibrium states of certain quantum and classical lattice, and continuum systems in two space dimensions.Partially supported by the N.S.F. under grant MCS 7801433.Partially supported by the N.S.F. under grant MCS 7906633.     

Recurrent random walks and the absence of continuous symmetry breaking on graphs

We consider geometrically disordered systems with a continuous symmetry groupG, where the internal degrees of freedom are attached to the vertices of a graph. We show that equilibrium states remainG-invariant at any temperatureT>0 if a random walk on the graph is recurrent. This generalizes a previous result obtained by Cassi.     

Thermodynamics of rotating self-gravitating systems

We investigate the statistical equilibrium properties of a system of classical particles interacting via Newtonian gravity, enclosed in a three-dimensional spherical volume. Within a mean-field approximation, we derive an equation for the density profiles maximizing the microcanonical entropy and solve it numerically. At low angular momenta, i.e. for a slowly rotating system, the well-known gravitational collapse “transition” is recovered. At higher angular momenta, instead, rotational symmetry can spontaneously break down giving rise to more complex equilibrium configurations, such as double-clusters (“double stars”). We analyze the thermodynamics of the system and the stability of the different equilibrium configurations against rotational symmetry breaking, and provide the global phase diagram. Received 8 July 2002 Published online 15 October 2002 RID="a" ID="a"e-mail: demartino@hmi.de