scholarly journals Concavity, Response Functions and Replica Energy

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 907 ◽  
Author(s):  
Alessandro Campa ◽  
Lapo Casetti ◽  
Ivan Latella ◽  
Agustín Pérez-Madrid ◽  
Stefano Ruffo
Keyword(s):  
Partition Function ◽  
Long Range ◽  
Thermodynamic Function ◽  
Thermodynamic Limit ◽  
Relevant Information ◽  
Negative Response ◽  
Response Functions ◽  
Thermodynamic Potentials ◽  
Interacting Systems ◽  
Number Of Particles

In nonadditive systems, like small systems or like long-range interacting systems even in the thermodynamic limit, ensemble inequivalence can be related to the occurrence of negative response functions, this in turn being connected with anomalous concavity properties of the thermodynamic potentials associated with the various ensembles. We show how the type and number of negative response functions depend on which of the quantities E, V and N (energy, volume and number of particles) are constrained in the ensemble. In particular, we consider the unconstrained ensemble in which E, V and N fluctuate, which is physically meaningful only for nonadditive systems. In fact, its partition function is associated with the replica energy, a thermodynamic function that identically vanishes when additivity holds, but that contains relevant information in nonadditive systems.

scholarly journals Relaxation of quasi-stationary states in long range interacting systems and a classification of the range of pair interactions

Open Physics ◽  
2012 ◽  
Vol 10 (3) ◽  
Author(s):  
Bruno Marcos ◽  
Andrea Gabrielli ◽  
Michael Joyce
Keyword(s):  
Long Range ◽  
Asymptotic Form ◽  
Pair Interaction ◽  
Collision Integral ◽  
Stationary States ◽  
Long Range Interactions ◽  
Interacting Systems ◽  
Dynamical Properties ◽  
Number Of Particles

AbstractSystems of particles interacting with long range interactions present generically ”quasi-stationary states” (QSS), which are approximately time-independent out of equilibrium states. In this proceedings, we explore the generalization of the formation of such QSS and their relaxation from the much studied case of gravity to a generic pair interaction with the asymptotic form of the potential v(r) ∼ 1/r γ with γ > 0 in d dimensions. We compute analytic estimations of the relaxation time calculating the rate of two body collisionality in a virialized system approximated as homogeneous. We show that for γ < (d − 1/2), the collision integral is dominated by the size of the system, while for γ > (d − 1/2), it is dominated by small impact parameters. In addition, the lifetime of QSS increases with the number of particles if γ < d − 1 (i.e. the force is not integrable) and decreases if γ > d − 1. Using numerical simulations we confirm our analytic results. A corollary of our work gives a ”dynamical” classification of interactions: the dynamical properties of the system depend on whether the pair force is integrable or not.

scholarly journals Effects of energy extensivity on the quantum phases of long-range interacting systems

2021 ◽  
Vol 103 (15) ◽  
Author(s):  
T. Botzung ◽  
D. Hagenmüller ◽  
G. Masella ◽  
J. Dubail ◽  
N. Defenu ◽  
...  
Keyword(s):  
Long Range ◽  
Quantum Phases ◽  
Interacting Systems

THE EXACT SOLUTION OF SOME LATTICE STATISTICS MODELS WITH FOUR STATES PER SITE

1964 ◽  
Vol 42 (8) ◽  
pp. 1564-1572 ◽  
Author(s):  
D. D. Betts
Keyword(s):  
Partition Function ◽  
Nearest Neighbor ◽  
Quaternary Alloy ◽  
Interesting Property ◽  
Two Dimensions ◽  
Lattice Statistics ◽  
Critical Temperatures ◽  
Different Types ◽  
Interacting Systems ◽  
Statistical Mechanical

Statistical mechanical ensembles of interacting systems localized at the sites of a regular lattice and each having four possible states are considered. A set of lattice functions is introduced which permits a considerable simplification of the partition function for general nearest-neighbor interactions. The particular case of the Potts four-state ferromagnet model is solved exactly in two dimensions. The order–disorder problem for a certain quaternary alloy model is also solved exactly on a square net. The quaternary alloy model has the interesting property that it has two critical temperatures and exhibits two different types of long-range order. The partition function for the spin-3/2 Ising model on a square net is expressed in terms of graphs without odd vertices, but has not been solved exactly.

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