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arXiv2009

Impurity-induced magnetic order in low dimensional spin gapped materials

We have studied the effect of non-magnetic Zn impurities in the coupled spin-ladder Bi(Cu1xZnx)2PO6 using 31P NMR, muSR and Quantum Monte Carlo simulations. Our results show that the impurities induce in their vicinity antiferromagnetic polarizations, extending over a few unit cells. At low temperature, these extended moments freeze in a process which is found universal among various other spin-gapped compounds: isolated ladders, Haldane or Spin-Peierls chains. This allows us to propose a simple common framework to explain the generic low-temperature impurity induced freezings observed in low dimensional spin-gapped materials.

cond-mat.str-el↗
arXiv2009

A simple mathematical proof of Boltzmann's equal a priori probability hypothesis

Using the Dissipation Theorem and a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we give a first-principles derivation of Boltzmann's postulate of equal a priori probability in phase space for the microcanonical ensemble. We show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.

cond-mat.stat-mech↗
arXiv2009

Molecular random tilings as glasses

We have recently shown [Blunt et al., Science 322, 1077 (2008)] that p-terphenyl-3,5,3',5'-tetracarboxylic acid adsorbed on graphite self-assembles into a two-dimensional rhombus random tiling. This tiling is close to ideal, displaying long range correlations punctuated by sparse localised tiling defects. In this paper we explore the analogy between dynamic arrest in this type of random tilings and that of structural glasses. We show that the structural relaxation of these systems is via the propagation--reaction of tiling defects, giving rise to dynamic heterogeneity. We study the scaling properties of the dynamics, and discuss connections with kinetically constrained models of glasses.

cond-mat.stat-mech↗
arXiv2009

Phase transitions and random quantum satisfiability

Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. We study random ensembles of the QMA$_1$-complete quantum satisfiability (QSAT) problem introduced by Bravyi. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for any hypergraph exhibit the same dimension of the satisfying manifold. This establishes the QSAT decision problem as equivalent to a, potentially new, graph theoretic problem and that the hardest typical instances are likely to be localized in a bounded range of clause density.

quant-ph↗
arXiv2009

Simple Model of First-Order Phase Transition

One-dimensional model of a system where first-order phase transition occurs is examined in the present paper. It is shown that basic properties of the phenomenon, such as a well defined temperature of transition, are caused both by existence of a border between the phases and the fact that only in the vicinity of that border it is possible for molecules to change their phase. Not only the model is introduced and theoretical analysis of its properties is made but also the results of Monte Carlo simulations are presented together with the results of numerical calculation of the distribution of energy levels of the system.

cond-mat.stat-mech↗
arXiv2010

Self-dual planar hypergraphs and exact bond percolation thresholds

Recent research on percolation has led to the construction of an infinite class of lattices for which the percolation thresholds can be determined exactly. We discuss the mathematical basis for the solutions of bond percolation models, and, in particular, attempt to address some misconceptions that might arise from the seminal articles by Ziff and Scullard. As one consequence, we show that there exist infinitely many real numbers $a \in [0,1]$ for which there are infinitely many lattices that have bond percolation threshold equal to $a$.

cond-mat.stat-mech↗
arXiv2009

Entropy production theorems and some consequences

The total entropy production fluctuations are studied in some exactly solvable models. For these systems, the detailed fluctuation theorem holds even in the transient state, provided initially the system is prepared in thermal equilibrium. The nature of entropy production during the relaxation of a system to equilibrium is analyzed. The averaged entropy production over a finite time interval gives a better bound for the average work performed on the system than that obtained from the well known Jarzynski equality. Moreover, the average entropy production as a quantifier for information theoretic nature of irreversibility for finite time nonequilibrium processes is discussed.

cond-mat.stat-mech↗
arXiv2009

Exact solution of the mixed spin-1/2 and spin-S Ising-Heisenberg diamond chain

The geometric frustration in a class of the mixed spin-1/2 and spin-S Ising-Heisenberg diamond chains is investigated by combining three exact analytical techniques: Kambe projection method, decoration-iteration transformation and transfer-matrix method. The ground state, the magnetization process and the specific heat as a function of the external magnetic field are particularly examined for different strengths of the geometric frustration. It is shown that the increase of the Heisenberg spin value S raises the number of intermediate magnetization plateaux, which emerge in magnetization curves provided that the ground state is highly degenerate on behalf of a sufficiently strong geometric frustration. On the other hand, all intermediate magnetization plateaux merge into a linear magnetization versus magnetic field dependence in the limit of classical Heisenberg spin S -> infinity. The enhanced magnetocaloric effect with cooling rate exceeding the one of paramagnetic salts is also detected when the disordered frustrated phase constitutes the ground state and the external magnetic field is small enough.

cond-mat.stat-mech↗
arXiv2009

Explosive growth in biased dynamic percolation on two-dimensional regular lattice networks

The growth of two-dimensional lattice bond percolation clusters through a cooperative Achlioptas-type of process, where the choice of which bond to occupy next depends upon the masses of the clusters it connects, is shown to go through an explosive, first-order kinetic phase transition with a sharp jump in the mass of the largest cluster as the number of bonds is increased. The critical behavior of this growth model is shown to be of a different universality class than standard percolation.

cond-mat.dis-nn↗
arXiv2009

Studying Soft Matter with "Soft" Potentials: Fast Lattice Monte Carlo Simulations and Corresponding Lattice Self-Consistent Field Calculations

The basic idea of fast Monte Carlo (MC) simulations is to perform particle-based MC simulations with the excluded-volume interactions modeled by "soft" repulsive potentials that allow particle overlapping. This gives much faster system relaxation and better sampling of the configurational space than conventional molecular simulations with "hard" repulsions that prevent particle overlapping. Here we present fast lattice MC (FLMC) simulations for confined homopolymers, where multiple occupancy of lattice sites is allowed with a proper Boltzmann weight and thus the evaluation of nearest-neighbor interactions can also be avoided. When compared with the corresponding lattice field theories based on the \emph{same} Hamiltonian, FLMC simulations further provide a powerful means for unambiguously and quantitatively revealing the correlation/fluctuation effects.

cond-mat.soft↗
arXiv2011

A first-passage-time theory for search and capture of chromosomes by microtubules in mitosis

The mitotic spindle is an important intermediate structure in eukaryotic cell division, in which each of a pair of duplicated chromosomes is attached through microtubules to centrosomal bodies located close to the two poles of the dividing cell. Several mechanisms are at work towards the formation of the spindle, one of which is the `capture' of chromosome pairs, held together by kinetochores, by randomly searching microtubules. Although the entire cell cycle can be up to 24 hours long, the mitotic phase typically takes only less than an hour. How does the cell keep the duration of mitosis within this limit? Previous theoretical studies have suggested that the chromosome search and capture is optimized by tuning the microtubule dynamic parameters to minimize the search time. In this paper, we examine this conjecture. We compute the mean search time for a single target by microtubules from a single nucleating site, using a systematic and rigorous theoretical approach, for arbitrary kinetic parameters. The result is extended to multiple targets and nucleating sites by physical arguments. Estimates of mitotic time scales are then obtained for different cells using experimental data. In yeast and mammalian cells, the observed changes in microtubule kinetics between interphase and mitosis are beneficial in reducing the search time. In {\it Xenopus} extracts, by contrast, the opposite effect is observed, in agreement with the current understanding that large cells use additional mechanisms to regulate the duration of the mitotic phase.

q-bio.SC↗
arXiv2009

A combinatorial solution for the current fluctuations in the exclusion process

We conjecture an exact expression for the large deviation function of the stationary state current in the partially asymmetric exclusion process with periodic boundary conditions. This expression is checked for small systems using functional Bethe Ansatz. It generalizes a previous result by Derrida and Lebowitz for the totally asymmetric exclusion process, and gives the known values for the three first cumulants of the current in the partially asymmetric model. Our result is written in terms of tree structures and provides a new example of a link between integrable models and combinatorics.

cond-mat.stat-mech↗
arXiv2009

Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: Exact and interpolation approaches

Generalized Bose-Einstein (BE) and Fermi-Dirac (FD) distributions in nonextensive quantum statistics have been discussed by the maximum-entropy method (MEM) with the optimum Lagrange multiplier based on the exact integral representation [Rajagopal, Mendes, and Lenzi, Phys. Rev. Lett. {\bf 80}, 3907 (1998)]. It has been shown that the $(q-1)$ expansion in the exact approach agrees with the result obtained by the asymptotic approach valid for $O(q-1)$. Model calculations have been made with a uniform density of states for electrons and with the Debye model for phonons. Based on the result of the exact approach, we have proposed the {\it interpolation approximation} to the generalized distributions, which yields results in agreement with the exact approach within $O(q-1)$ and in high- and low-temperature limits. By using the four methods of the exact, interpolation, factorization and superstatistical approaches, we have calculated coefficients in the generalized Sommerfeld expansion, and electronic and phonon specific heats at low temperatures. A comparison among the four methods has shown that the interpolation approximation is potentially useful in the nonextensive quantum statistics. Supplementary discussions have been made on the $(q-1)$ expansion of the generalized distributions based on the exact approach with the use of the un-normalized MEM, whose results also agree with those of the asymptotic approach.

cond-mat.stat-mech↗
arXiv2009

Typical case behaviour of spin systems in random graph and composite ensembles

This thesis includes analysis of disordered spin ensembles corresponding to Exact Cover, a multi-access channel problem, and composite models combining sparse and dense interactions. The satisfiability problem in Exact Cover is addressed using a statistical analysis of a simple branch and bound algorithm. The algorithm can be formulated in the large system limit as a branching process, for which critical properties can be analysed. Far from the critical point a set of differential equations may be used to model the process, and these are solved by numerical integration and exact bounding methods. The multi-access channel problem is formulated as an equilibrium statistical physics problem for the case of bit transmission on a channel with power control and synchronisation. A sparse code division multiple access method is considered and the optimal detection properties are examined in typical case by use of the replica method, and compared to detection performance achieved by iterative decoding methods. These codes are found to have phenomena closely resembling the well-understood dense codes. The composite model is introduced as an abstraction of canonical sparse and dense disordered spin models. The model includes couplings due to both dense and sparse topologies simultaneously. Through an exact replica analysis at high temperature, and variational approaches at low temperature, several phenomena uncharacteristic of either sparse or dense models are demonstrated. An extension of the composite interaction structure to a code division multiple access method is presented. The new type of codes are shown to outperform sparse and dense codes in some regimes both in optimal performance, and in performance achieved by iterative detection methods in finite systems.

cond-mat.dis-nn↗
arXiv2012

A first step phenomenology for the statistics of non-equilibrium fluctuations

The paper assesses stationary probability distributions in out of equilibrium systems. In the phenomenology proposed, no free energy can be well defined. Fluctuations of Landau free energy couplings arise when the intrinsic chemical potential leads to intrinsic disorder. The relaxation is shown to take the form of a geometrical random process. Systems of this kind show criticality features as well as that of first order transitions, which encapsulate in the form of a generalized static fluctuation dissipation relation. This will help determine three classes of distributions, which are, by defining a Hurst exponent for the relaxation rate: the regular Maxwell-Boltzmann for all H < 1/2; the usual scale free universal distributions with power law tails for H in ]1/2,1]; and a new class. The latter lies in the intermediate case, when H = 1/2. The distribution functions are scale free close to the origin, and cross-over to a Maxwell-Boltzmann asymptotic behavior, with both the scaling exponent and an effective temperature that depend on the magnitude of the intrinsic disorder.

cond-mat.stat-mech↗
arXiv2012

Thermodynamic Casimir Effect in the large-n limit

We consider systems with slab geometry of finite thickness L that undergo second order phase transitions in the bulk limit and belong to the universality class of O(n)-symmetric systems with short-range interactions. In these systems the critical fluctuations at the bulk critical temperature T_c induce a long-range effective force called the "thermodynamic Casimir force". We describe the systems in the framework of the O(n)-symmetric phi^4-model, restricting us to the large-n limit n->infty. In this limit the physically relevant case of three space dimensions d=3 can be treated analytically in systems with translational symmetry as, e.g., in the bulk or slabs with periodic or antiperiodic boundary conditions. We consider Dirichlet and open boundary conditions at the surfaces that break the translational invariance along the axis perpendicular to the slab. From the broken translational invariance we conclude the necessity to solve the systems numerically. We evaluate the Casimir amplitudes for Dirichlet and open boundary conditions on both surfaces and for Dirichlet on one and open on the other surface. Belonging to the same surface universality class we find the expected asymptotic equivalence of Dirichlet and open boundary conditions. To test the quality of our method we confirm the analytical results for periodic and antiperiodic boundary conditions.

cond-mat.stat-mech↗
arXiv2009

Casimir Force Between Dielectric Media with Free Charges

The statistical mechanical approach to Casimir problems for dielectrics separated by a vacuum gap turns out to be compact and effective. A central ingredient of this method is the effect of interacting fluctuating dipole moments of the polarizable particles. At arbitrary temperature the path integral formulation of quantized particles, developed by Høye-Stell and others, is needed. At high temperature - the limit considered in the present paper - the classical theory is however sufficient. Our present theory is related to an idea put forward earlier by Jancovici and {Š}amaj (2004), namely to evaluate the Casimir force between parallel plates invoking an electronic plasma model and the Debye-Hückel theory for electrolytes. Their result was recently recovered by Høye (2008), using a related statistical mechanical method. In the present paper we generalize this by including a constant permittivity in the description. The present paper generalizes our earlier theory for parallel plates (1998), as well as for spherical dielectrics (2001). We also consider the Casimir force between a polarizable particle and a conductor with a small density of charges, finding agreement with the result recently derived by Pitaevskii (2008).

quant-ph↗
arXiv2009

Entanglement in Far From Equilibrium Stationary States

We present four estimators of the entanglement (or interdepency) of ground-states in which the coefficients are all real nonnegative and therefore can be interpreted as probabilities of configurations. Such ground-states of hermitian and non-hermitian Hamiltonians can be given, for example, by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models. We consider in detail the last case. Using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive, conformal invariant and non-conformal invariant massless phases. We show that like in the case of the quantum problem, the estimators verify the area law and can therefore be used to signal phase transitions in stationary states.

cond-mat.stat-mech↗