An algebraic approach to discrete mechanics

Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.     

A statistical approach to quantum mechanics

A Monte Carlo method is used to evaluate the Euclidean version of Feynman's sum over particle histories. Following Feynman's treatment, individual paths are defined on a discrete (imaginary) time lattice with periodic boundary conditions. On each lattice site, a continuous position variable x_i specifies the spacial location of the particle. Using a modified Metropolis algorithm, the low-lying energy eigenvalues, |ψ₀(x)|², the propagator, and the effective potential for the anharmonic oscillator are computed, in good agreement with theory. For a deep double-well potential, instantons were found in our computer simulations appearing as multi-kink configurations on the lattice.     

Quantum approach to classical statistical mechanics

We present a new approach to study the thermodynamic properties of d-dimensional classical systems by reducing the problem to the computation of ground state properties of a d-dimensional quantum model. This classical-to-quantum mapping allows us to extend the scope of standard optimization methods by unifying them under a general framework. The quantum annealing method is naturally extended to simulate classical systems at finite temperatures. We derive the rates to assure convergence to the optimal thermodynamic state using the adiabatic theorem of quantum mechanics. For simulated and quantum annealing, we obtain the asymptotic rates of T(t) approximately (pN)/(k(B)logt) and gamma(t) approximately (Nt)(-c/N), for the temperature and magnetic field, respectively. Other annealing strategies are also discussed.     

A statistical mechanics approach to Granovetter theory

In this paper we try to bridge breakthroughs in quantitative sociology/econometrics, pioneered during the last decades by Mac Fadden, Brock–Durlauf, Granovetter and Watts–Strogatz, by introducing a minimal model able to reproduce essentially all the features of social behavior highlighted by these authors.     

A differential algebraic approach to mechanics with perfect nonholonomic constraints

Differential algebra permits the generalization of several features of symplectic formalism to mechanics with perfect nonholonomic constraints.     

A new statistical mechanics approach to dense granular media

A new statistical mechanics approach to dense granular media is presented. The thermodynamic formalism is set out directly in terms of elastic potential energy, such that the configurational temperature (the intensive property which defines the steady state) relates to a quadratic function of the stresses (rather than other linear functions used in recent developments). Dense granular media are considered as canonical ensembles of noninteracting clusters, which can be identified with repeatable equilibrium configurations. Then, particles can be located in a new proposed phase space (conceived to separate the elastic potential energy levels). Although the importance of this paper lies in the method itself, it has been illustratively applied to the simple case of two-dimensional (2D) dense granular media (an arrangement of frictionless monodisperse elastic disks under isotropic horizontal stress compression). In this case, the temperature is directly replaced by the squared external pressure, and the packing ratio of the most probable microstate is close to the reported value of random close packing. Moreover, some interesting general conclusions arise.     

On non-linear non-equilibrium statistical mechanics

The exact equations of motion for slow variables, which characterize a macroscopic system far from equilibrium, are derived with the help of a new time-dependent projection operator. The present formalism is a natural extension of Mori's approach to the non-linear region far from equilibrium, or of Kawasaki-Gunton's approach to quantal systems.     

On the algebraic structure of quantum mechanics

We present a reformulation of the axiomatic basis of quantum mechanics with particular reference to the manner in which the usual algebraic structures arise from certain natural physical requirements. Care is taken to distinguish between features of physical significance and those introduced for mathematical convenience. Our conclusion is that the usual algebraic structures cannot be significantly generalised without conflicting with our current experimental picture of processes occurring at the quantum level.     

New microscopic approach to the statistical mechanics of chemical association

We consider a new approach to the statistical mechanics of chemical association, A+BAB. In principle it is exact, and its thermodynamic basis goes back to Gibbs, but its statistical mechanical implementation in terms of molecular models does not appear to have been given before. For practical computations the success of the method hinges on our ability to calculate the free energy for a mixture of A, B, and AB species, where the species concentrations are regarded as independent. We illustrate the method by analysis of some simple limiting cases.     

A molecular quantum mechanics approach to evaluate molecular properties in liquid phase using statistical mechanics

A method for the calculation of molecular properties in liquid phase is proposed. The theoretical development links in a natural fashion Molecular Quantum Mechanics and Statistical Mechanics. The method can be applied to molecular liquids in thermodynamical equilibrium without strong interactions between molecules. After the Introduction (in which some fundamental ideas are discussed) the model is developed in its simplest form, giving a numerical application of interest for IR spectroscopy in liquid phase. Immediatly, this preliminar version is generalized for pure liquids taking into account the polarization effects on all the molecules of the sample. A numerical example on carbon disulphide shows the convergence of this method and its sensitivity to the structural changes in the liquid. In the last section, some future improvements are suggested and several topics related with the study of molecular liquids are considered.     

Land cover change using an energy transition paradigm in a statistical mechanics approach

This paper explores a statistical mechanics approach as a means to better understand specific land cover changes on a continental scale. Integrated assessment models are used to calculate the impact of anthropogenic emissions via the coupling of technoeconomic and earth/atmospheric system models and they have often overlooked or oversimplified the evolution of land cover change. Different time scales and the uncertainties inherent in long term projections of land cover make their coupling to integrated assessment models difficult. The mainstream approach to land cover modelling is rule-based methodology and this necessarily implies that decision mechanisms are often removed from the physical geospatial realities, therefore a number of questions remain: How much of the predictive power of land cover change can be linked to the physical situation as opposed to social and policy realities? Can land cover change be understood using a statistical approach that includes only economic drivers and the availability of resources? In this paper, we use an energy transition paradigm as a means to predict this change. A cost function is applied to developed land covers for urban and agricultural areas. The counting of area is addressed using specific examples of a Pólya process involving Maxwell–Boltzmann and Bose–Einstein statistics. We apply an iterative counting method and compare the simulated statistics with fractional land cover data with a multi-national database. An energy level paradigm is used as a basis in a flow model for land cover change. The model is compared with tabulated land cover change in Europe for the period 1990–2000. The model post-predicts changes for each nation. When strong extraneous factors are absent, the model shows promise in reproducing data and can provide a means to test hypothesis for the standard rules-based algorithms.     

A unified approach to phase diagrams in field theory and statistical mechanics

We construct the phase diagram of any system which admits a low-temperature polymer or cluster expansion. Such an expansion turns the system into a hard-core interacting contour model with small, but not necessarily positive, activities. The method uses some of Zahradnik's ideas [Z1], but applies equally well to systems with complex interactions. We give two applications. First, to low-temperatureP()₂ models with complex couplings; and second, to a computation of asymptotics of partition functions in periodic volumes. If the index of a supersymmetric field theory is known, the second application would help determine the number of phases in infinite volume.Alfred P. Solan Research Fellow. Supported in part by the National Science Foundation under Grants PHY87-064220, DMS 88-58073, and PHY/DMS 86-45122     

Significance analysis and statistical mechanics: an application to clustering

This Letter addresses the statistical significance of structures in random data: given a set of vectors and a measure of mutual similarity, how likely is it that a subset of these vectors forms a cluster with enhanced similarity among its elements? The computation of this cluster p value for randomly distributed vectors is mapped onto a well-defined problem of statistical mechanics. We solve this problem analytically, establishing a connection between the physics of quenched disorder and multiple-testing statistics in clustering and related problems. In an application to gene expression data, we find a remarkable link between the statistical significance of a cluster and the functional relationships between its genes.     

On the inverse problem in statistical mechanics

For quantum spin systems it is known that for a suitable space of potentials the equilibrium states areW*-dense in the set of all translation invariant states. The problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential. A necessary and sufficient condition for a state to be an equilibrium state for some potential is given in Sect. 3.     

On the inverse problem in statistical mechanics

We review briefly the status of some inverse problems in classical equilibrium statistical mechanics.This paper is dedicated to Professor P. G. Bergmann on the occasion of his sixtieth birthday.Supported by NSF grant No. GP42614X.     

On the statistical mechanics of rigid particles

In this paper, generalized functions are used in the configuration partition function for fluids composed of arbitrarily shaped, rigid particles. This leads to new expressions for the basic statistical thermodynamic functions and some equations that may be useful in developing approximate theories, such as the scaled particle theory, for such fluids. The results are applicable to a large class of arbitrarily shaped, rigid particles and reduce exactly to the usual hard-sphere expressions.