[en] In this note, we will discuss how to compactly express the Jarzynski identity for an open quantum system with dissipative dynamics. In quantum dynamics we must avoid explicitly measuring the work directly, which is tantamount to continuously monitoring the state of the system, and instead measure the heat flow from the environment. These measurements can be concisely represented with Hermitian map superoperators, which provide a convenient and compact representations of correlation functions and sequential measurements of quantum systems

[en] We study fluctuations of pressure in equilibrium for classical particle systems. In equilibrium statistical mechanics, pressure for a microscopic state is defined by the derivative of a thermodynamic function or, more mechanically, through the momentum current. We show that although the two expectation values converge to the same equilibrium value in the thermodynamic limit, the variance of the mechanical pressure is in general greater than that of the pressure defined through the thermodynamic relation. We also present a condition for experimentally detecting the difference between them in an idealized measurement of momentum transfer.

[en] In this paper, we modify the Langevin dynamics associated to the generalized Curie–Weiss model by introducing noisy and dissipative evolution in the interaction potential. We show that, when a zero-mean Gaussian is taken as single-site distribution, the dynamics in the thermodynamic limit can be described by a finite set of ODEs. Depending on the form of the interaction function, the system can have several phase transitions at different critical temperatures. Because of the dissipation effect, not only the magnetization of the systems displays a self-sustained periodic behavior at sufficiently low temperature, but, in certain regimes, any (finite) number of stable limit cycles can exist. We explore some of these peculiarities with explicit examples.

[en] We investigate dynamics of a supersymmetric fermion lattice model introduced by Nicolai (J Phys A 9:1497–1505, 1976). We show that the Nicolai model has infinitely many local constants of motion for its Heisenberg time evolution, and therefore ergodicity (with respect to thermal equilibrium states) breaks. It has infinitely many degenerated classical ground states. This phenomena is considered as localization at zero temperature. From a viewpoint of perturbation theory, we explain why delocalization is suppressed at zero temperature despite its disorder-free translation-invariant quantum interaction.

[en] Hard core bosons in a large class of one or two dimensional flat band systems have an upper critical density, below which the ground states can be described completely. At the critical density, the ground states are Wigner crystals. If one adds a particle to the system at the critical density, the ground state and the low lying multi particle states of the system can be described as a Wigner crystal with an additional pair of particles. The energy band for the pair is separated from the rest of the multi-particle spectrum. The proofs use a Gerschgorin type of argument for block diagonally dominant matrices. In certain one-dimensional or tree-like structures one can show that the pair is localised, for example in the chequerboard chain. For this one-dimensional system with periodic boundary condition the energy band for the pair is flat, the pair is localised.

[en] The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas.

[en] Consider the following coverage model on $\mathbb{N}$, for each site $i\in <!--\; ???\; -->\mathbb{N}$ associate a pair $({\mathrm{\xi <!--\; ??\; -->}}_i,R_i)$ where $({\mathrm{\xi <!--\; ??\; -->}}_i{)}_{i\ge <!--\; ???\; -->0}$ is a 1-dimensional undelayed discrete renewal point process and $(R_i{)}_{i\ge <!--\; ???\; -->0}$ is an i.i.d. sequence of $\mathbb{N}$-valued random variables. At each site where ${\mathrm{\xi <!--\; ??\; -->}}_i=1$ start an interval of length $R_i$. Coverage occurs if every site of $\mathbb{N}$ is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.

[en] We consider the two-dimensional Ising model with long-range pair interactions of the form $J_{xy}\sim <!--\; ???\; -->|x-<!--\; ???\; -->y{|}^{-<!--\; ???\; -->\mathrm{\alpha <!--\; ??\; -->}}$ with $\mathrm{\alpha <!--\; ??\; -->}>2$, mostly when $J_{xy}\ge <!--\; ???\; -->0$. We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed ± boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman–Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts.

[en] We consider bond percolation on ${\mathbb{Z}}^d\times <!--\; ??\; -->{\mathbb{Z}}^s$ where edges of ${\mathbb{Z}}^d$ are open with probability $p<p_c({\mathbb{Z}}^d)$ and edges of ${\mathbb{Z}}^s$ are open with probability q, independently of all others. We obtain bounds for the critical curve in (p, q), with p close to the critical threshold $p_c({\mathbb{Z}}^d)$. The results are related to the so-called dimensional crossover from ${\mathbb{Z}}^d$ to ${\mathbb{Z}}^{d+s}$.

[en] We study the set of p-adic Gibbs measures of the q-state Potts model on the Cayley tree of order three. We prove the vastness of the set of the periodic p-adic Gibbs measures for such model by showing the chaotic behavior of the corresponding Potts–Bethe mapping over ${\mathbb{Q}}_p$ for the prime numbers $p\equiv <!--\; ???\; -->1(\mathrm{m}\mathrm{o}\mathrm{d}3)$. In fact, for $0<|\mathrm{\theta <!--\; ??\; -->}-<!--\; ???\; -->1{|}_p<|q{|}_p^2<1$ where $\mathrm{\theta <!--\; ??\; -->}={\exp }_p<!--\; ???\; -->(J)$ and J is a coupling constant, there exists a subsystem that is isometrically conjugate to the full shift on three symbols. Meanwhile, for $0<|q{|}_p^2\le <!--\; ???\; -->|\mathrm{\theta <!--\; ??\; -->}-<!--\; ???\; -->1{|}_p<|q{|}_p<1$, there exists a subsystem that is isometrically conjugate to a subshift of finite type on r symbols where $r\ge <!--\; ???\; -->4$. However, these subshifts on r symbols are all topologically conjugate to the full shift on three symbols. The p-adic Gibbs measures of the same model for the prime numbers $p=2,3$ and the corresponding Potts–Bethe mapping are also discussed. On the other hand, for $0<|\mathrm{\theta <!--\; ??\; -->}-<!--\; ???\; -->1{|}_p<|q{|}_p<1,$ we remark that the Potts–Bethe mapping is not chaotic when $p=3$ and $p\equiv <!--\; ???\; -->2(\mathrm{m}\mathrm{o}\mathrm{d}3)$ and we could not conclude the vastness of the set of the periodic p-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case $0<|q{|}_p\le <!--\; ???\; -->|\mathrm{\theta <!--\; ??\; -->}-<!--\; ???\; -->1{|}_p<1$ for all prime numbers p.