Search results for " Statistical"
showing 10 items of 1649 documents
Elastic moduli, dislocation core energy and melting of hard disks in two dimensions
2000
Elastic moduli and dislocation core energy of the triangular solid of hard disks of diameter $\sigma$ are obtained in the limit of vanishing dislocation- antidislocation pair density, from Monte Carlo simulations which incorporates a constraint, namely that all moves altering the local connectivity away from that of the ideal triangular lattice are rejected. In this limit, we show that the solid is stable against all other fluctuations at least upto densities as low as $\rho \sigma^2 = 0.88$. Our system does not show any phase transition so diverging correlation lengths leading to finite size effects and slow relaxations do not exist. The dislocation pair formation probability is estimated …
Corner contribution to cluster numbers in the Potts model
2013
For the two-dimensional Q-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number N_Gamma of clusters that intersect a given contour Gamma. To leading order, N_Gamma is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of Gamma. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are no…
Monte Carlo Methods: a powerful tool of statistical physics
1998
Statistical mechanics of condensed matter systems (solids, fluids) tries to express macroscopic equilibrium properties of matter as averages computed from a Hamiltonian that expresses interactions of an atomistic many body system. While analytic methods for most problems involve crude and uncontrolled approximations, the Monte Carlo computer simulation method allows a numerically exact treatment of this problem, apart from “statistical errors” which can be made as small as desired, and the systematic problem that a system of finite size is treated rather than the thermodynamic limit. However, the simulations of phase transitions then elucidate how a symmetry breaking arises via breaking of …
Two topologically distinct Dirac-line semimetal phases and topological phase transitions in rhombohedrally stacked honeycomb lattices
2018
Three-dimensional topological semimetals can support band crossings along one-dimensional curves in the momentum space (nodal lines or Dirac lines) protected by structural symmetries and topology. We consider rhombohedrally (ABC) stacked honeycomb lattices supporting Dirac lines protected by time-reversal, inversion and spin rotation symmetries. For typical band structure parameters there exists a pair of nodal lines in the momentum space extending through the whole Brillouin zone in the stacking direction. We show that these Dirac lines are topologically distinct from the usual Dirac lines which form closed loops inside the Brillouin zone. In particular, an energy gap can be opened only by…
Rounding of Phase Transitions in Cylindrical Pores
2010
Phase transitions of systems confined in long cylindrical pores (capillary condensation, wetting, crystallization, etc.) are intrinsically not sharply defined but rounded. The finite size of the cross section causes destruction of long range order along the pore axis by spontaneous nucleation of domain walls. This rounding is analyzed for two models (Ising/lattice gas and Asakura-Oosawa model for colloid-polymer mixtures) by Monte Carlo simulations and interpreted by a phenomenological theory. We show that characteristic differences between the behavior of pores of finite length and infinitely long pores occur. In pores of finite length a rounded transition occurs first, from phase coexiste…
Fluids in extreme confinement.
2012
For extremely confined fluids with two-dimensional density $n$ in slit geometry of accessible width $L$, we prove that in the limit $L\to 0$ the lateral and transversal degrees of freedom decouple, and the latter become ideal-gas-like. For small wall separation the transverse degrees of freedom can be integrated out and renormalize the interaction potential. We identify $n L^2 $ as hidden smallness parameter of the confinement problem and evaluate the effective two-body potential analytically, which allows calculating the leading correction to the free energy exactly. Explicitly, we map a fluid of hard spheres in extreme confinement onto a 2d-fluid of disks with an effective hard-core diame…
Dynamical mean-field theory and weakly non-linear analysis for the phase separation of active Brownian particles
2015
Recently, we have derived an effective Cahn-Hilliard equation for the phase separation dynamics of active Brownian particles by performing a weakly non-linear analysis of the effective hydrodynamic equations for density and polarization [Speck et al., Phys. Rev. Lett. 112, 218304 (2014)]. Here, we develop and explore this strategy in more detail and show explicitly how to get to such a large-scale, mean-field description starting from the microscopic dynamics. The effective free energy emerging from this approach has the form of a conventional Ginzburg-Landau function. On the coarsest scale, our results thus agree with the mapping of active phase separation onto that of passive fluids with …
Second-Order Phase Transition Induced by Deterministic Fluctuations in Aperiodic Eight-State Potts Models
1999
We investigate the influence of aperiodic modulations of the exchange interactions between nearest-neighbour rows on the phase transition of the two-dimensional eight-state Potts model. The systems are studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm for different aperiodic sequences. The transition point is located through duality relations, and the critical behaviour is investigated using FSS techniques at criticality. While the pure system exhibits a first-order transition, we show that the deterministic fluctuations resulting from the aperiodic coupling distribution are liable to modify drastically the physical properties in the nei…
Spectral energy distribution and generalized Wien's law for photons and cosmic string loops
2014
Physical objects with energy $u_w(l) \sim l^{-3w}$ with $l$ characteristic length and $w$ a dimensionless constant, lead to an equation of state $p=w\rho$, with $p$ the pressure and $\rho$ the energy density. Special entities with thisbproperty are, for instance, photons ($u = hc/l$, with $l$ the wavelength) with $w = 1/3$, and some models of cosmic string loops ($u =(c^4/aG)l$, with $l$ the length of the loop and $a$ a numerical constant), with $w = -1/3$. Here, we discuss some features of the spectral energy distribution of these systems and the corresponding generalization of Wien's law, which in terms of $l$ has the form $Tl_{mp}^{3w}=constant$, being $l_{mp}$ the most probable size of …
Trajectory Statistics of Confined L\'evy Flights and Boltzmann-type Equilibria
2013
We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise, where Langevin representation is absent. In view of the L\'{e}vy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Here, we infer pdf $\rho (x,t)$ based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for $\rho (x,t)$ and its target pdf $\rho_*(x)$. To simulate the above processes, we construct a suitable modification of t…