We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian Mean Field (HMF) model as a simple example. These systems generically undergo a violent relaxation to a quasi-stationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasi-stationary solution of the Vlasov equation, slowly evolving in time due to finite $N$ effects. For subcritical energies $7/12<U<3/4$, we exhibit cases where the DF is well-fitted by a Tsallis $q$-distribution with an index $q(t)$ slowly decreasing in time from $q\simeq 3$ (semi-ellipse) to $q=1$ (Boltzmann). When the index $q(t)$ reaches a critical value $q_{crit}(U)$, the non-magnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energies, we report for the first time the existence of a magnetized QSS with a very long lifetime.
Comment on ``Nonextensive hamiltonian systems follow Boltzmann's principle not Tsallis statistics-phase transition, second law of thermodynamics'' by Gross
Nonadditive Tsallis $q$-statistics has successfully been applied for a plethora of systems in natural sciences and other branches of knowledge. Nevertheless, its foundations have been severely criticised by some authors based on the standard additive Boltzmann-Gibbs approach thereby remaining a quite controversial subject. In order to clarify some polemical concepts, the distribution function for an ideal gas with a finite number of point particles and its $q$-index are analytically determined. The two-particle correlation function is also derived. The degree of correlation diminishes continuo…
Temporal extensivity of Tsallis' entropy and the bound on entropy production rate
Tsallis熵是Boltzmann-Gibbs熵的推广，在复杂系统的非广延统计力学中起着核心作用。近年来，人们在建立Tsallis熵的动力学基础方面做了大量的工作。它们主要研究处于混沌边缘的非线性动力系统。在这里，它是通过推广的热统计公式的基础上的时间平均最近提出的Carati [A。Carati，Physica A 348，110（2005）]，只要相关，由$q$索引的Tsallis熵在时间上是广泛的：时间上的线性增长，即，有限熵产生率然后，熵产生率上的普遍界被证明是$1/|1-q|$。相关概率过程的属性，即，分析了在相空间中决定运动随机性的逗留时间分布。
We prove a proposition that the entropy of the system composed of finite $N$ molecules of ideal gas is the $q$-entropy or Havrda-Charvát-Tsallis entropy, which is also known as Tsallis entropy, with the entropic index $q=\frac{D(N-1)-4}{D(N-1)-2}$ in $D$-dimensional space. The indispensable infinity assumption used by Boltzmann and others in their derivation of entropy formulae is not involved in our derivation, therefore our derived formula is exact. The analogy of the $N$-body system brings us to obtain the entropic index of a combined system $q_C$ formed from subsystems having different ent…
Following Chakrabarti,Chandrasekhar, and Naina [Physica A {\bf 389} (2010) 1571], we attempt a classical relativistic treatment of Verlinde's emergent entropic force conjecture by appealing to a relativistic Hamiltonian in the context of Tsalli's statistics. The ensuing partition function becomes the classical one for small velocities. We show that Tsallis' relativistic (classical) free particle distribution at temperature $T$ can generate Newton's gravitational force's $r^{-2}$ {\it distance's dependence}. If we want to repeat the concomitant argument by appealing to Renyi's distribution, the…
Non-extensive thermodynamics and stationary processes of localization
我们把注意力集中在熵指数Q<1的动力学过程上。根据Tsallis和Bukman的概率论[C.Tsallis，D. J. Bukman，Phys.Rev.E54，R2197（1996）]，这些过程本质上是亚扩散的。非广泛的推广Kolmogorov-Sinai熵产生相同的熵指数意味着平稳条件。我们注意到，另一方面，强制执行的子扩散的固定属性产生的本地化过程发生在有限的时间尺度内的效果。因此，我们得出结论，Q<1的平稳动力学过程必须经历发生在有限时间的局部化过程。我们检查这个结论的有效性，通过数值处理的动力学的Logistic映射在临界点。
Rényi and Tsallis information entropies for a harmonic position-dependent mass
University of Education, Winneba（温纳巴教育学院）；Ecole Polytechnique d’Abomey Calavi (EPAC-UAC)（阿博美-卡拉维高等理工学院）；Université d’Abomey-Calavi (UAC)（阿博美-卡拉维大学）；Institut de Mathématiques et de Sciences Physiques (IMSP)（数学与物理科学研究所）；University of Ghana（加纳大学）；African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究所）
We introduce a ${\mathbb Z}_2$-index for time reversal invariant Hamiltonians with unique gapped ground state on quantum spin chains. We show this is an invariant of a $C^1$-classification of gapped Hamiltonians.
