B. Simon, The Statistical Mechanics of Lattice Gases, Vol. 1, Princeton University Press, Princeton NJ (1993).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of algebraic quantum statistical mechanics which appears to be particularly well suited to the study of general structural properties of non equilibrium steady states. The basic notions of this algebraic framework are briefly introduced in Section 2. The reader is referred to the monographs [BR1, BR2, Ha, Si1, Th] for more detailed expositions. In Section 3, we define non equilibrium steady states and discuss their basic structural properties. Section 4 is devoted to the notion of entropy production. Finally, a simple class of models with strictly positive entropy production is described in Section 5. ....

....exist at least one (and possibly many) KMS states on O for any 2 R. Such states are constructed as thermodynamic limits of local KMS states defined on O . Under some additional regularity conditions on the interaction the KMS state is unique for small enough . We refer the reader to [BR2, I, Ru1, Si1] for detailed information on the kinematical structure and equilibrium thermodynamics of quantum spin systems, and to [Ru3, Ru4, Ru5, Ru6] for their non equilibrium statistical mechanics. The dynamical aspects of quantum spin systems are comparatively little studied. To our knowledge the only ....

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Simon, B.: Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton, N.J., (1993).

....physicists in the last 2030 years and a lot of interesting facts have been rigorously established. We proceed by listing the properties of the two dimensional model which will ultimately be needed in this paper. For general overviews of various aspects mentioned below we refer to, e.g. [12, 28, 29, 51]. The readers familiar with the background (and the standard notation) should feel free to skip the remainder of this section and go directly to Section 1.3 where we discuss the main results of the present paper. Bulk properties. For all # 0, the measure P has a unique infinite volume ....

B. Simon, The Statistical Mechanics of Lattice Gases, Vol. I., Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993.

....prove that Theorem 1.1 (p) is an analytic function of the complex variable p around p = 1. Our proof of the above theorem is based upon a polymer gas representation for 1 (p) The polymer expansion is a standard technique in equilibrium statistical mechanics and quantum eld theory (see e.g. [1, 4, 6]) and its has been applied with success to many di erent problems. Although a polymer expansion for the percolation model for p 0 is immediate (in this case, lattice animals play the role of polymers) it is not clear how to do it for p 1. As a matter of fact, in this regime the statistical ....

....be rewritten as ( 1 ; n )2( 1 ) n ) e 1 i j n ; 3:11) where ( is the n times cartesian product of . It is thus a standard task in cluster expansion theory to compute explicitly the Mayer expansion for the function ln ; see e.g. 2] [6]) Doing such a calculation one obtains ln ; 1 ; n ) 1 ) n ) 3:12) where the Ursell coecients ( 1 ; n ) are given by ( 1 ; n ) 1) if n 2 1 if n = 1. 3:13) The sum in (3.13) ....

Simon B.: The Statistical Mechanics of Lattice Gases, Volume 1, Princeton University Press (1993) 13

....of algebraic quantum statistical mechanics which appears to be particularly well suited to the study of general structural properties of non equilibrium steady states. The basic notions of this algebraic framework are briefly introduced in Section 2. The reader is referred to the monographs [Ha, BR1, BR2, Si1, Th] for more detailed expositions. In Section 3, we define non equilibrium steady states and discuss their basic structural properties. Section 4 is devoted to the notion of entropy production. Finally, a simple class of models with strictly positive entropy production is described in Section 5. This ....

....exist at least one (and possibly many) KMS states on O for any 2 R. Such states are constructed as thermodynamic limits of local KMS states defined on O . Under some additional regularity conditions on the interaction the KMS state is unique for small enough . We refer the reader to [Ru1, I, BR2, Si1] for detailed information on the kinematical structure and equilibrium thermodynamics of quantum spin systems, and to [Ru3, Ru4, Ru5, Ru6] for their non equilibrium statistical mechanics. The dynamical aspects of quantum systems are comparatively little studied. To our knowledge the only ....

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Simon, B.: Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton, N.J., (1993).

....for short range systems, a mathematical foundation of this belief has not been presented in a general context. A number of rigorous results have related various lattice systems to their mean field counterparts, either in the form of bounds on transition temperatures and critical exponents, see [19, 20, 51] and references therein, or in terms of limits of the free energy [47] and the magnetization [12, 40] as the dimension tends to infinity. In all of these results, the nature of the phase transition is not addressed or the proofs require special symmetries which, as it turns out, ensure that the ....

....b) if there are at least two distinct infinite volume limits of the measure in (1.2) arising from different boundary conditions. We will call these limiting objects either infinite volume Gibbs measures or, in accordance with mathematical physics nomenclature, Gibbs states. We refer the reader to [26,51] for more details on the general properties of Gibbs states and phase transitions. We remark that, while the entire class of models has been written so as to appear identical, the physics will be quite different depending on the particulars of# and , and the inner product. Indeed, the language ....

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B. Simon, The statistical mechanics of lattice gases, Vol. I., Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993.

....overcome this deficiency one can consider even more simplified models such as lattice systems. There, thanks to the Pirogov Sinai [15] 22] theory, some exactly solvable models, inequalities, etc. one has much better control over the low temperature region of the phase diagram [17] 20] 7] [19]. Alternatively, one can consider, following the pioneering work of van der Waals [21] 18] 16] 5] mean field theories (MFT) yielding approximate equations for state or free energies which indeed exhibit most of the qualitative and many quantitative features of real world phase diagrams. ....

Simon B., The Statistical Mechanics of Lattice Gases, Vol I. Princeton University Press, Princeton, New Jersey (1993).

....a Gibbs state on the space time configurations. For the moment, we forget how this Gibbsian description is obtained but we will come back to this in a remark below. For the sake of simplicity let us in fact consider the case where we are dealing with a Gibbs state for a lattice spin system, see [20, 11, 38] for background. The configurations oe are then elements of Omega = G where G, the single spin space, is finite and oe = oe(n; i) n 2 ; i 2 ) is a space time trajectory for n = discrete time and i a spatial coordinate on the d Gammadimensional lattice . Physically, it is better to ....

Simon, B. (1993) The Statistical Mechanics of Lattice Gases, Volume 1, Princeton University Press, Princeton.

....discrete symmetry breaking. Here we formulate the Pirogov Sinai theory focusing on linear functionals that are tangent to the free energy. This allows both to avoid the difficulties associated with boundary conditions, and to make the link with the notion of KMS states, using well known results [Ara, Isr, Sim]. We reformulate results of [BKU, DFF, DFFR, KU] in this framework, and extend the theory to a class of models where discrete symmetries are broken at intermediate temperatures. This applies in particular to some systems with continuous symmetries. For this, we consider the restricted ensembles ....

....Finally, Section 6 contains proofs of our main result by studying a contour model and deriving the required bounds on the contours. 2. General framework of quantum lattice models 2.1. Basic set up. We consider a quantum mechanical system on a dimensional lattice Z , as considered e.g. in [Rue, Isr, BR, Sim]. We will need a slight modification of the usual formalism, in order to treat fermionic lattice gases [DFFR] and in particular to accommodate the fact that fermionic creation and annihilation operators do not commute but anticommute. A quantum lattice system is defined by the following data: ....

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B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press (1993)

....the reader to the relevant literature; moreover, our style will be mostly heuristic and nonmathematical, with some remarks added for the mathematically inclined reader. 2 Gibbs States Since there exist many good references on the theory of Gibbs measures (also called Gibbs states) see e.g. [24, 47, 57, 88, 87]) we shall only state the main de nition and the basic properties. To start with a concrete example, consider the nearest neighbour Ising model on Z d . To each i 2 Z d , we associate a variable i 2 f1; 1g, and the (formal) Hamiltonian is H = J X hiji ( i j 1) 2.1) where hiji ....

B. Simon. The Statistical Mechanics of Lattice Gases, Princeton University Press, Princeton, 1993.

....is devoted to the computation via cluster expansion of the spin spin correlations exploiting (0. 3) 2 This is in some sense quite standard (see the references above) however our careful treatment of the convergence of the series, using a representation of the spin spin correlation borrowed from [Si] and the BattleBrydges Federbush trees technique, allows us to obtain the above mentioned decay of the correlations with the same power as the interaction. The representation of the spin spin correlation is presented in section 4. In section 5 we state our main general result and we add to it ....

....JC(J; i) where C(J; i) is the constant appearing in the estimate (3.9) is the basic tool needed to obtain such convergence. However, since such convergence is a byproduct of the convergence of the spin spin truncated correlations (1. 7) we give here their explicit expression in term of polymers [Si], and we treat directly the problem of the convergence of the correlations. This is achieved just recalling lemma 1 and noting that the following identity holds ( x1 ; x2 ) k ff 1 ff 2 log Z ( ff 1 ; ff 2 ) fi fi fi ff=0 (4:5) where Z ( ff 1 ; ff 2 ) Z d ....

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Simon, B.: The statistical mechanics of lattice gases. Vol. I. Princeton Series in Physics. Princeton University Press, Princeton, NJ, 1993.

....infinite and the fugacities are nonnegative, Gibbs measures can then be defined by the usual Dobrushin Lanford Ruelle prescription [23] At small w we expect to have uniqueness of the Gibbs measure, exponential decay of correlations, analyticity, etc. Indeed, the Dobrushin uniqueness theorem [23, 55] easily implies the uniqueness of the infinite volume Gibbs measure, and the exponential decay of correlations in this unique Gibbs measure, whenever 0 w x (1 Gamma ffl) d x Gamma 1 ffl) for all vertices x [here ffl 0, and d x denotes the number of vertices adjacent to x] In particular, ....

....on the negative real axis. 4. Theorem 2.2 (which can be generalized to soft core repulsive interactions [57] provides an extraordinarily simple proof of the convergence of the Mayer expansion for such lattice gases. Recall that the usual approach to proving convergence of the Mayer expansion [42, 53, 13, 8, 55, 9, 60] is to explicitly bound the expansion coefficients; this requires some rather nontrivial combinatorics (for example, an inequality of Penrose [42] together with the counting of trees) Once this is done, an immediate consequence is that Z is nonvanishing in any polydisc where the series for log Z ....

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Simon, B. (1993) The Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton NJ.

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B. Simon, The Statistical Mechanics of Lattice Gases. Vol. I, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993.

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B. Simon, The Statistical Mechanics of Lattice Gases, Vol. 1, Princeton University Press, Princeton NJ (1993).

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Barry Simon. The statistical mechanics of lattice gases. Vol. I. Princeton Series in Physics. Princeton University Press, Princeton, NJ, 1993.

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B. Simon, The Statistical Mechanics of Lattice Gases (Princeton University Press, Princeton, NJ, 1993).

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Simon, B. (1993) The Statistical Mechanics of Lattice Gases, Vol. I, Princeton, N.J.: Princeton University Press.

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B. SIMON, The statistical mechanics of lattice gases, Vol. I, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993.

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Simon, B. (1993) The Statistical Mechanics of Lattice Gases, Volume 1, Princeton University Press, Princeton.

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B. Simon, The Statistical Mechanics of Lattice Gases, Princeton University Press, Princeton, 1993. 19

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B.Simon, The statistical mechanics of lattice gases. Princeton Univ. Press. (1992) 13

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Simon, B.: Statistical Mechanics of Lattice Gases. Princeton University Press, Princeton, N.J., (1993).

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B. Simon, \The statistical mechanics of lattice gases." Vol. I. Princeton University Press, Princeton, NJ, USA, 1993.

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B. Simon, The Statistical Mechanics of Lattice Gases (Princeton University Press, Princeton, 1993).

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B. Simon. The Statistical Mechanics of Lattice Gases, Princeton University Press, Princeton, 1993.

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B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press (1993).

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