Yang-Lee Distribution of Zeros for a van der Waals Gas

Hemmer, P. C.; Hauge, E. Hiis

doi:10.1103/physrev.133.a1010

Cited by 40 publications

(15 citation statements)

References 7 publications

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“…As we have already mentioned, there are some other models, for which the Lee-Yang theorem does not hold [42][43][44][45][46][47][48][49][50][51][52][53]. Unlike the above examples, the problem we have considered in this paper concerns ferromagnetic Ising model, and the Lee-Yang theorem was proven [55] to hold for any Ising-like model with ferromagnetic interaction, see also [56].…”

Section: Discussionmentioning

confidence: 99%

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

Krasnytska

Berche

Holovatch

et al. 2016

J. Phys. A: Math. Theor.

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We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as P (k) ∼ k −λ . We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case λ > 5, reproduces the zeros for the Ising model on a complete graph. For 3 < λ < 5 we derive the λ-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the λ-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when 3 < λ < 5. Whereas in the former case the zeros are purely imaginary, they have a non zero real part in latter case, so that the celebrated Lee-Yang circle theorem is violated.

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Section: Discussionmentioning

confidence: 99%

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

Krasnytska

Berche

Holovatch

et al. 2016

J. Phys. A: Math. Theor.

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“…In this context the Yang-Lee circle theorem states that the zeros of the partition function occur on the unit circle |z| = 1 in the complex plane of the fugacity, z = e βµ ′ (where µ ′ is chemical potential) and, in the thermodynamic limit, form an arc of this unit circle extending from θ = π over to complex-conjugate endpoints at z = e ±iθe [4]. For closed-form approximations to the equation of state of a liquid-gas system such as that of van der Waals, the density of Yang-Lee zeros has been calculated [35,37]. For intermolecular potentials with a repulsive hard core, the expansion of the reduced pressure pv 0 /(k B T ) in powers of fugacity, pv 0 /(k B T ) = ∞ j=1 b ℓ z ℓ , exhibits alternating signs, indicating a singularity on the negative real z axis [36].…”

Section: Connection Of Singular Behavior Of the Zero Density In The µ...mentioning

confidence: 99%

On properties of the Ising model for complex energy/temperature and magnetic field

Matveev

Shrock

2008

J. Phys. A: Math. Theor.

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We study some properties of the Ising model in the plane of the complex (energy/temperature)dependent variable u = e −4K , where K = J/(kBT ), for nonzero external magnetic field, H. Exact results are given for the phase diagram in the u plane for the model in one dimension and on infinitelength quasi-one-dimensional strips. In the case of real h = H/(kBT ), these results provide new insights into features of our earlier study of this case. We also consider complex h = H/(kBT ) and µ = e −2h . Calculations of complex-u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary h, i.e., µ = e iθ , we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the u plane. We find that in this case, the phase boundary Bu contains a real line segment extending through part of the physical ferromagnetic interval 0 ≤ u ≤ 1, with a right-hand endpoint u rhe at the temperature for which the Yang-Lee edge singularity occurs at µ = e ±iθ . Conformal field theory arguments are used to relate the singularities at u rhe and the Yang-Lee edge.

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“…This is what is currently called the inverse problem (in analogy with the inverse problem in electrostatics), and it has been addressed in several papers on various specific systems, see Refs. 20, 21 (two models of Ising ferromagnets), 22 (a Tonks gas of hard rods and a gas with a weak long-range repulsion), 31 (Tonks gas), 23,24,27,28,29,30 (the van der Waals gas), 25 (a lattice gas with a hard-core repulsion extended over several lattice sites), 32, 33 (Takahashi lattice gas), 34 (point particles with logarithmic interactions), and even in an experimental setup, see Ref. 35 (for a two-dimensional Ising ferromagnet).…”

Section: Grand-canonical Partition Function Zeros and The Equation Ofmentioning

confidence: 99%

Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang–lee Formalism

Bena

Droz

Lipowski

2005

Int. J. Mod. Phys. B

164

147

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Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang-Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions. YANG-LEE FORMALISM FOR PHASE TRANSITIONS 5particle separations, i.e., in terms of the interaction potential, |u(r)| C 1 /r d+ε as r → ∞ (with C 1 > 0 and ε positive constants), where d is the dimension of the system. Such systems are currently called short-range interaction systems. (ii) u(r) has to have a repulsive part increasing sufficiently rapid at small interparticle distances (preventing the system from collapse at high particle number densities). Throughout this paper, we shall consider systems with a hard-core interparticle repulsion at small distances, u(r) = ∞ for r < r 0 ∼ b 1/d (where b is the single-particle excluded volume), which means that a system of volume V can accommodate a finite maximum number of particles M = V /b. c (iii) Finally, the interaction potential has to be everywhere bounded from below, u(r) −u 0 whatever r (with u 0 a positive constant).Note a rather general result concerning the one-dimensional equilibrium systems with short range interactions, namely van Hove's theorem 1,5,7 , according to which no phase transition is possible in such systems, in contrast to what happens generically in corresponding nonequilibrium systems, see Ref. 8 for a brief review. A recent critical discussion 9 of this result allowed to highlight some exceptions to this theorem, but we shall not be concerned here with these rather pathological situations.Many physical situations can be modeled by interaction potentials with the characteristics (i) -(iii) above. There are, however, a few exceptions. The most salient example is that of the systems with long-range interactions (or non-additive systems), which include, e.g., systems with gravitational or Coulombian forces (see Ref. 6 for a modern review of their (thermo)dynamical properties studies). Such systems exhibit an unequivalence of the statistical ensembles, and it is not clear yet how to apply the concepts of the Yang-Lee theory (as described below) to their phase transitions. We will not address them further here. A limiting case of such systems is that of mean-field interactions, on which we will briefly comment in Secs. 7.2, 12, 13.3. Yang-Lee zeros of the grand-canonical partition function: the general frameworkWe shall address here two problems, namely:(A) The location of the phase transition point: How, when knowing the partition function (in the thermodynamic limit), can one locate a phase transition point by investigating the zeros of this partition function with respect to the control parameter of the system.(B) The characteristics of the transition: How can one extract informa...

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Yang-Lee Distribution of Zeros for a van der Waals Gas

Cited by 40 publications

References 7 publications

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

On properties of the Ising model for complex energy/temperature and magnetic field

Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang–lee Formalism

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