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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 39 (1974), S. 165-184 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ϱ, and for the spin correlation functions σ, are essentially equivalent for all ϱ, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.
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  • 3
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. We study, globally in time, the velocity distribution f(v,t) of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field E. The density f satisfies a Boltzmann-type kinetic equation containing a fully nonlinear electron‐electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the L 1 distance between f and a certain time-dependent Maxwellian stays small uniformly in t. Moreover, the mean and variance of this time‐dependent Maxwellian satisfy a coupled set of nonlinear ordinary differential equations that constitute the “hydrodynamical” equations for this kinetic system. This remains true even when these ordinary differential equations have non‐unique equilibria, thus proving the existence of multiple stable stationary solutions for the full kinetic model. Our approach relies on scale‐independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globally in time.
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  • 4
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(−2mHβ) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsn s (x 1, ...,x s ; β,z) are analytic in the two variables β andH if the reciprocal temperature β〉0 and the complex numberH is not a limit point of zeros of the grand partition function ξ, and (ii) the Ursell functionsu s (x 1, ...,x s ; β,z) tend to 0 as Δ s ≡Max i, j |x i −x j | → ∞ if β〉0 and ReH≠0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value λ, then |u s |〈C exp [(−2 βm |ReH|+ε) Δ s /λ] where ε is any small positive number andC is independent of Δ s . One consequence of the result (i) is that a phase transition can occur as β is varied at constantH only ifH is a limit point of zeros of ξ (which can happen only if ReH=0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition whenH is varied at constant β. For a lattice or continuum gas with non-negative interaction potential (corresponding, in the lattice case, to an Ising antiferromagnet), similar results are shown to hold provided β〉0 and the complex fugacityz is less than the radius of convergence of the Mayerz expansion; for the continuum gas, however,n s andu s must be replaced by their values integrated over small volumes surrounding each of the pointsx 2, ...,x s . It is shown that the pressurep is analytic in both β andz, if it is analytic inz at fixed β over a suitable range of values of β andz, and further that, except for continuum systems without hard cores,p,n s andu s have convergent Maclaurin expansions in β for small enoughz.
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  • 5
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We generalize the notion of “ground states” in the Pirogov-Sinai theory of first order phase transitions at low temperatures, applicable to lattice systems with a finite number of periodic ground states to that of “restricted ensembles” with equal free energies. A restricted ensemble is a Gibbs ensemble, i.e. equilibrium probability measure, on a restricted set of configurations in the phase space of the system. When a restricted ensemble contains only one configuration it coincides with a ground state. In the more general case the entropy is also important. An example of a system we can treat by our methods is theq-state Potts model where we prove that forq sufficiently large there exists a temperature at which the system coexists inq+1 phases;q-ordered phases are small modifications of theq perfectly ordered ground states and one disordered phase which is a modification of the restricted ensemble consisting of all “perfectly disordered” (neighboring sites must have different spins) configurations. The free energy thus consists entirely of energy in the firstq-restricted ensembles and of entropy in the last one. Our main motivation for this work is to develop a rigorous theory for phase transitions in continuum fluids in which there is no symmetry between the phases, e.g. the liquid-vapour phase transition. The present work goes a certain way in that direction.
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  • 6
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A ν-dimensional classical particle system in a torus, i.e., in a rectangular box with periodic boundary conditions, is considered in a canonical ensemble. Subject to mild restrictions over and above the usual stability and tempering conditions it is proved that the thermodynamic limit for the torus exists and is identical with that for systems contained in normal domains with boundaries or walls. If, in addition, the pair interaction potential ϕ(r) decreases sufficiently rapidly (so thatr∣ϕ(r)∣ is integrable at ∞), and satisfies some further regularity conditions, then the difference between the free energies of the torus and of the corresponding box is at most of the order of a surface term. Somewhat stronger results are indicated for the grand canonical pressure.
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  • 7
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider general ferromagnetic spin systems with finite range interactions and an even single-spin distribution of compact support on IR. It is shown under mild assumptions on the single-spin distribution that a low temperature expansion, in powers ofT, for the free energy and the correlation functions is asymptotic. We also prove exponential clustering in the pure phases and analyticity of the free energy and of the correlation functions in the reciprocal temperature β for Re β large.
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  • 8
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider a one-dimensional model of a system in contact with a heat bath: A particle (the system ormolecule) of massM, confined to the unit interval [0, 1], is surrounded by an infinite ideal gas (thebath of atoms) of point particles of massm with which it interacts via elastic collisions. The atoms are not affected by the walls at 0 and at 1. We obtain “convergence to equilibrium” for the molecule, from essentially any initial distribution on its position and velocity. The infinite composite system of molecule and bath has very good ergodic properties: it is a Bernoulli system.
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  • 9
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
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  • 10
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The one to one correspondence between the existence of a unique equilibrium state and the differentiability of the free energy density with respect to the external field previously shown for Ising ferromagnetis is extendend to higher valued spin systems as well as to continuum systems satisfying the Fortuin, Kasteleyn and Ginibre inequalities. In particular this is shown to hold for a mixture ofA –B particles in which there is no interaction between like particles and a repulsion between unlike particles. Where the derivative of the free energy is discontinuous there are at least two equilibrium states.
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