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A tutorial introduction to the statistical mechanics of phase transitions and phase coexistence is presented, starting out from equilibrium systems and nonequilibrium steady-state situations in ext...

he contact angle of a liquid droplet on a surface under partial wetting conditions differs for a nanoscopically rough or periodically corrugated surface from its value for a perfectly flat surface. Wenzel's relation attributes this difference simply to the geometric magnification of the surface area (by a factor $r_{\rm w}$), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form $z_{\rm wall}(y) = \Delta\cos(2\pi y/\lambda)$ , for a potential of short range $\sigma_{\rm w}$ acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change of the wall-vapor surface tension can be computed exactly, and corrections to Wenzel's equation are typically of order $\sigma_{\rm w}\Delta/\lambda^2$. For fixed $r_{\rm w}$ and fixed $\sigma_{\rm w}$ the approach to Wenzel's result with increasing $\lambda$ may be nonmonotonic and this limit often is only reached for $\lambda/\sigma_{\rm w}>30$. For a non-additive binary mixture, density functional theory is used to work out the density profiles of both coexisting phases both for planar and corrugated walls, as well as the corresponding surface tensions. Again, deviations from Wenzel's results of similar magnitude as in the above ideal gas case are predicted. Finally, a crudely simplified description based on the interface Hamiltonian concept is used to interpret corresponding simulation results along similar lines. Wenzel's approach is found to generally hold when $\lambda/\sigma_{\rm w}\gg 1$, $\Delta/\lambda

Semiflexible polymers bound to planar substrates by a short-range surface potential are studied by Molecular Dynamics simulations to clarify the extent to which these chain molecules can be considered as strictly two-dimensional. Applying a coarse-grained bead-spring model, the chain length N and stiffness &kappa, as well as the strength of the adsorption potential ϵ w a l l are varied over a wide range. The excluded-volume (EV) interactions inherent in this model can also be &ldquo, switched off&rdquo, to provide a discretized version of the Kratky&ndash, Porod wormlike chain model. We study both local order parameters (fraction f of monomers within the range of the potential, bond-orientational order parameter &eta, ) and the mean square gyration radius parallel, &lang, R g 2 &rang, | , and perpendicular, &lang, &perp, to the wall. While for strongly adsorbed chains EV has negligible effect on f and &eta, we find that &lang, | is strongly affected when the chain contour length exceeds the persistence length. Monomer coordinates in perpendicular (&perp, ) direction are correlated over the scale of the deflection length which is estimated. It is found that f , &eta, and &lang, converge to their asymptotic values with 1 / N corrections. For both weakly and strongly adsorbed chains, the distribution functions of &ldquo, loops&rdquo, &ldquo, trains&rdquo, and &ldquo, tails&rdquo, are analyzed. Some consequences pertaining to the analysis of experiments on adsorbed semiflexible polymers are pointed out.

In a theoretical description of homogeneous nucleation one frequently assumes an "equilibrium" coexistence of a liquid droplet with surrounding vapor of a density exceeding that of a saturated vapor at bulk vapor-liquid two-phase coexistence. Thereby one ignores the caveat that in the thermodynamic limit, for which the vapor would be called supersaturated, such states will at best be metastable with finite lifetime, and thus not be well-defined within equilibrium statistical mechanics. In contrast, in a system of finite volume stable equilibrium coexistence of droplet and supersaturated vapor at constant total density is perfectly possible, and numerical analysis of equilibrium free energies of finite systems allows to obtain physically relevant results. In particular, such an analysis can be used to derive the dependence of the droplet surface tension γ( R) on the droplet radius R by computer simulations. Unfortunately, however, the precision of the results produced by this approach turns out to be seriously affected by a hitherto unexplained spurious dependence of γ( R) on the total volume V of the simulation box. These finite size effects are studied here for the standard Ising/lattice gas model in d = 2 dimensions and an Ising model on the face-centered cubic lattice with 3-spin interaction, lacking symmetry between vapor and liquid phases. There also the analogous case of bubbles surrounded by undersaturated liquid is treated. It is argued that (at least a large part of) the finite size effects result from the translation entropy of the droplet or bubble in the system. This effect has been shown earlier to occur also for planar interfaces for simulations in the slab geometry. Consequences for the estimation of the Tolman length are briefly discussed. In particular, we find clear evidence that in d = 2 the leading correction of the curvature-dependent interface tension is a logarithmic term, compatible with theoretical expectations, and we show that then the standard Tolman-style analysis is inapplicable.

Conformations of partially or fully adsorbed semiflexible polymer chains are studied varying both contour length $L$, chain stiffness, $\ensuremath{\kappa}$, and the strength of the adsorption potential over a wide range. Molecular dynamics simulations show that partially adsorbed chains (with ``tails,'' surface attached ``trains,'' and ``loops'') are not described by the Kratky-Porod wormlike chain model. The crossover of the persistence length from its three-dimensional value (${\ensuremath{\ell}}_{p}$) to the enhanced value in two dimensions ($2{\ensuremath{\ell}}_{p}$) is analyzed, and excluded volume effects are identified for $L\ensuremath{\gg}{\ensuremath{\ell}}_{p}$. Consequences for the interpretation of experiments are suggested. We verify the prediction that the adsorption threshold scales as ${\ensuremath{\ell}}_{p}^{\ensuremath{-}1/3}$.

Finite-size scaling for a first-order phase transition where a continuous symmetry is broken is developed using an approximation of Gaussian probability distributions with a phenomenological ``degeneracy'' factor included. Predictions are compared with data from Monte Carlo simulations of the three-dimensional, $XXZ$ Heisenberg antiferromagnet in a field in order to study the finite-size behavior on a $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattice for the first-order ``spin-flop'' transition between the Ising-like antiferromagnetic state and the canted, $XY$-like state. Our theory predicts that for large linear dimension $L$ the field dependence of all moments of the order parameters as well as the fourth-order cumulants exhibit universal intersections. Corrections to leading order should scale as the inverse volume. The values of these intersections at the spin-flop transition point can be expressed in terms of a factor $q$ that characterizes the relative degeneracy of the ordered phases. Our theory yields $q=\ensuremath{\pi}$, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality can be invoked for first-order phase transitions.

To be specific, let us consider for the moment the problem of N atoms in a volume V at temperature T, and we wish to calculate the average of some observable A which in quantum mechanics is described by an operator Â.

Roughly at the time (1987) when the manuscript for the first three chapters of the present book was completed, several breakthroughs occurred. They had a profound influence on the scope of Monte Carlo simulations in statistical physics, particularly for the study of phase transitions in lattice models.

A great triumph of statistical physics in the latter part of the 20th century was the understanding of critical behavior and universality at second-order phase transitions. In contrast, first-order transitions were believed to have no common features. However, we argue that the classic, first-order “spin-flop” transition (between the antiferromagnetic and the rotationally degenerate, canted state) in an anisotropic antiferromagnet in a magnetic field exhibits a new kind of universality. We present a finite-size scaling theory for a first-order phase transition where a continuous symmetry is broken using an approximation of Gaussian probability distributions with a phenomenological degeneracy factor “q” included, where “q” characterizes the relative degeneracy of the ordered phases. Predictions are compared with high resolution Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field to study the finite-size behavior for L×L×L simple cubic lattices. The field dependence of all moments of the order parameters exhibit universal intersections at the spin-flop transition. Our Monte Carlo data agree with theoretical predictions for asymptotic large L behavior. Our theory yields q = π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality.

We study active Brownian particles as a paradigm for a genuine nonequilibrium phase transition requiring steady driving. Access to the critical point in computer simulations is obstructed by the fact that the density is conserved. We propose a method based on arguments from finite-size scaling to determine critical points and successfully test it for the two-dimensional (2D) Ising model. Using this method allows us to accurately determine the critical point of two-dimensional active Brownian particles at ${\mathrm{Pe}}_{\text{cr}}=40(2), {\ensuremath{\phi}}_{\text{cr}}=0.597(3)$. Based on this estimate, we study the corresponding critical exponents $\ensuremath{\beta}, \ensuremath{\gamma}/\ensuremath{\nu}$, and $\ensuremath{\nu}$. Our results are incompatible with the 2D-Ising exponents, thus raising the question whether there exists a corresponding nonequilibrium universality class.

Monte Carlo simulations of crystal nuclei coexisting with the fluid phase in thermal equilibrium in finite volumes are presented and analyzed, for fluid densities from dense melts to the vapor. Generalizing the lever-rule for two-phase coexistence in the canonical ensemble to finite volume, "measurements" of the nucleus volume together with the pressure and chemical potential of the surrounding fluid allows to extract the surface free energy of the nucleus. Neither the knowledge of the (in general non-spherical) nucleus shape nor of the angle-dependent interface tension is required for this task. The feasibility of the approach is demonstrated for a variant of the Asakura-Oosawa model for colloid-polymer mixtures, which form face-centered cubic colloidal crystals. For a polymer to colloid size ratio of $0.15$, the colloid packing fraction in the fluid phase can be varied from melt values to zero by the variation of an effective attractive potential between the colloids. It is found that the approximation of spherical crystal nuclei often underestimates actual nucleation barriers significantly. Nucleation barriers are found to scale as $\Delta F^*=(4\pi/3)^{1/3}\bar{\gamma}(V^*)^{2/3}+const.$ with the nucleus volume $V^*$, and the effective surface tension $\bar{\gamma}$ that accounts implicitly for the nonspherical shape can be precisely estimated., Comment: 17 pages, 17 figures

We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields establishes that the average magnetization of the sample, with critical exponent β, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by γ+2β=ν +ν requires β=1/2 (γ=4, ν =3, and ν =2), the thermodynamic scaling establishes that Δs=γ+β, which in contrast requires β=0 (Δs=4), where γ, ν, ν, and Δs are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., β=0) and a direct measurement of the susceptibility critical exponent γ/ν =2.0±0.2, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise., Instituto de Física de Líquidos y Sistemas Biológicos

As a rule, mean-field theories applied to a fluid that can undergo a transition from saturated vapor at density ρυ to a liquid at density ρl yield a van der Waals loop. For example, isotherms of the chemical potential μ(T,ρ) as a function of the density ρ at a fixed temperature T less than the critical temperature Tc exhibit a maximum and a minimum. Metastable and unstable parts of the van der Waals loop can be eliminated by the Maxwell construction. Van der Waals loops and the corresponding double minimum potentials are mean-field artifacts. Simulations at fixed μ=μcoex for ρυ

The nematic ordering in semiflexible polymers with contour length $L$ exceeding their persistence length $\ell_p$ is described by a confinement of the polymers in a cylinder of radius $r_{eff}$ much larger than the radius $r_\rho$, expected from the respective concentration of the solution. Large scale Molecular Dynamics simulations combined with Density Functional Theory are used to locate the Isotropic-Nematic ($I-N$)-transition and to validate this cylindrical confinement. Anomalous fluctuations, due to chain deflections from neighboring chains in the nematic phase are proposed. Considering deflections as collective excitations in the nematically ordered phase of semiflexible polymers elucidates the origins of shortcomings in the description of the $I-N$ transition by existing theories., Comment: 5 pages, 5 figures

We study numerically the critical behavior of a modified, active Asakura-Oosawa model for colloid-polymer mixtures. The colloids are modeled as self-propelled particles with Vicsek-like interactions. This system undergoes phase separation between a colloid-rich and a polymer-rich phase, whereby the phase diagram depends on the strength of the Vicsek-like interactions. Employing a subsystem-block-density distribution analysis, we determine the critical point and make an attempt to estimate the critical exponents. In contrast to the passive model, we find that the critical point is not located on the rectilinear diameter. A first estimate of the critical exponents $\beta$ and $\nu$ is consistent with the underlying 3d-Ising universality class observed for the passive model., Comment: 9 pages, 8 figures

The two-dimensional Ising model in the geometry of a long stripe can be regarded as a model system for the study of nanopores. As a quasi-one-dimensional system, it also exhibits a rather interesting “phase behavior”: At low temperatures the stripe is either filled with “liquid” or “gas” and “densities” are similar to those in the bulk. When we approach a “pseudo-critical point” (below the critical point of the bulk) at which the correlation length becomes comparable to the length of the stripe, several interfaces emerge and the systems contains multiple “liquid” and “gas” domains. The transition depends on the size of the stripe and occurs at lower temperatures for larger stripes. Our results are corroborated by simulations of the three-dimensional Asakura–Oosawa model in cylindrical geometry, which displays qualitatively similar behavior. Thus our simulations explain the physical basis for the occurrence of “hysteresis critical points” in corresponding experiments.

The finite size scaling analysis of Monte Carlo data is discussed for two models for which hyperscaling is violated: (i) the random field Ising model (using a model for a colloid-polymer mixture in a random matrix as a representative) (ii) The Ising bi-pyramid in computing surface fields.

We present simulation results for the phase behavior of a single chain for a flexible lattice polymer model using the Wang–Landau sampling idea. Using the micro-canonical density of states obtained with this method we will discuss the ability of an analysis in the micro-canonical ensemble to locate the coil-globule (continuous) and liquid–solid (first-order) transitions found for this problem using a canonical analysis.

In this article we consider mixtures of non-adsorbing polymers and rod-like colloids in the isotropic phase, which upon the addition of polymers show an effective attraction via depletion forces. Above a certain concentration, the depletant causes phase separation of the mixture. We performed Monte Carlo simulations to estimate the phase boundaries of isotropic–isotropic coexistence. To determine the phase boundaries we simulated in the grand canonical ensemble using successive umbrella sampling [J. Chem. Phys. 120 (2004) 10925]. The location of the critical point was estimated by a finite size scaling analysis. In order to equilibrate the system efficiently, we used a cluster move in which rods and spheres were exchanged.

The interplay between surface and interface effects on binary AB mixtures that are confined in unconventional geometries is investigated by Monte Carlo simulations and phenomenological considerations. Both double-wedge and bi-pyramid confinements are considered and competing surface fields are applied at the two opposing halves of the system. Below the bulk critical temperature, domains of opposite order parameter are stabilized at the corresponding corners and an interface runs across the middle of the bi-partite geometry. Upon decreasing the temperature further one encounters a phase transition at which the AB symmetry is broken. The interface is localized in one of the two wedges or pyramids, respectively, and the order parameter is finite. In both cases, the transition becomes discontinuous in the thermodynamic limit but it is not a first-order phase transition. In an antisymmetric double wedge geometry the transition is closely related to the wedge-filling transition. Choosing the ratio of the cross-section L × L of the wedge and its length L y according to L y /L 3 = const., simulations and phenomenological consideration show that the new type of phase transition is characterized by critical exponents α = 3/4, β = 0, and γ = 5/4 for the specific heat, order parameter, and susceptibility, respectively. In an antisymmetric bi-pyramid the transition occurs at the cone-filling transition of a single pyramid. The important critical fluctuations are associated with the uniform translation of the interface and they can be described by a Landau-type free energy. Monte Carlo results provide evidence that the coefficients of this Landau-type free energy exhibit a system-size dependence, which gives rise to critical amplitudes that diverge with system size and result in a transition that becomes discontinuous in the thermodynamic limit.

The properties of Ising square lattices with nearest neighbor ferromagnetic exchange confined in a corner geometry, are studied by means of Monte Carlo simulations. Free boundary conditions at which boundary magnetic fields ± h are applied, i.e., at the two boundary rows ending at the lower left corner a field + h acts, while at the two boundary rows ending at the upper right corner a field − h acts. For temperatures T less than the critical temperature T c of the bulk, this boundary condition leads to the formation of two domains with opposite orientation of the magnetization direction, separated by an interface which for T larger than the filling transition temperature T f ( h ) runs from the upper left corner to the lower right corner, while for T T f ( h ) this interface is localized either close to the lower left corner or close to the upper right corner. It is shown that for T = T f ( h ) the magnetization profile m ( z ) in the z-direction normal to the interface simply is linear and the interfacial width scales as w ∝ L , while for T > T f ( h ) it scales as w ∝ L . The distribution P ( l ) of the interface position l (measured along the z-direction from the corners) decays exponentially for T T f ( h ) from either corner, is essentially flat for T = T f ( h ) , and is a Gaussian centered at the middle of the diagonal for T > T f ( h ) . Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.

solids to complex multi-component materials and even biological matter, are governed by well understood laws of physics: on the relevant scales of length and time; the appropriate description would be just the Schrodinger equation for the quantum-many-body problem of the nuclei and electrons interacting with Coulomb forces. Statistical mechanics would then provide the framework to extend this quantum theory of condensed matter towards a statistical description in terms of averages taken at nonzero temperature. However, this program cannot be carried out straightforwardly: even dealing only with the associated electrons is still a challenge for the methods of quantum chemistry. Similarly, standard statistical mechanics makes precise explicit statements only on the properties of systems for which the manybody problem can be effectively reduced to a problem of independent particles or quasi-particles. Such problems are, for instance, ideal gases, paramagnets, or the multidimensional harmonic oscillator describing phonons in harmonic crystals. While all these problems are useful and educative (we hence teach them all to students to illustrate the spirit of the general theoretical framework) they do not encompass most of the problems of interest in condensed matter physics: the interactions among the considered degrees of freedom introduce nontrivial correlations between them. Systematic perturbation-theory type methods usually do not lead very far, and uncontrolled closed form approximations often fail utterly. A well-known example is the study of the liquid gas transition in fluids: simple theories such as the van der Waals equation and its extensions (e.g. the "perturbed chain" statistical associating fluid theory [PC-SAFT] intended to model the equation of state of polymer melts [1]) give rise to spurious loops in the isotherms, fail to describe the critical behavior correctly, and may even predict completely unphysical phase equilibria that do not correspond to any physical behavior of the system but are mere artefacts of the inappropriate approximations [1]. The theory of condensed matter systems contains a never-ending list of such failures! Hence, until about 50 years ago, condensed matter theory suffered from the basic problem that only a formal framework for the theoretical description in terms of quantum theory and statistical mechanics did exist, while a reliable set of tools that would allow accurate explicit predictions to be made for the static and dynamic properties of these systems from first principles was simply missing! This unsatisfactory situation has changed fundamentally through the invention of computer simulation, which provides a much more promising approach to these problems. Computer simulation is a novel methodic route allowing the Condensed matter theory by computer simulation: from materials to chemical biology [DOI: 10.1051/EPN:2007009]

A two-dimensional magnetic Ising system confined in an L × D geometry ( L ⪡ D ) in the presence of competing magnetic fields (h) acting at opposite walls along the D -direction, exhibits an interface between domains of different orientation that run parallel to the walls. In the limit L → ∞ , this interface undergoes a wetting transition that occurs at the critical curve T w ( h ) , so that for T T w ( h ) such an interface is bound to the walls, while for T w ( h ) ⩽ T T cb the interface is freely fluctuating around the center of the film, where T cb is the bulk critical temperature. By considering both short- and long-range magnetic fields acting at the walls, we study the divergence of the (equilibrated) average position of the interface when approaching the wetting critical point. Furthermore, starting from a monodomain structure with the interface bound to one wall, we also study the dynamics of the interface unbinding.

A two-dimensional magnetic Ising system confined in an L × D geometry () in the presence of competing magnetic fields (h) acting at opposite walls along the D-direction exhibits an interface between domains of different orientation that runs parallel to the walls. In the limit of infinite film thickness () this interface undergoes a wetting transition that occurs at the critical curve Tw(h), so that for T

The nature of the melting transition for a system of hard disks with translational degrees of freedom in two spatial dimensions has been analysed by a combination of computer simulation methods and a finite size scaling technique. The behaviour of the system is consistent with the predictions of the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory. The structural and elastic properties of binary colloidal mixtures in two and three spatial dimensions are discussed as well as those of colloidal systems with quenched point impurities. Hard and soft disks in external periodic (light) fields show rich phase diagrams, including freezing and melting transitions when the density of the system is varied. Using Monte Carlo simulation methods we have investigated the phase diagrams of such systems for various values of the density and the amplitude of the external potential. The conductance quantization of atomic Au wires is discussed as well as the properties of Si clusters.

We present extensive Monte Carlo simulations of the Ising film confined in an L × M geometry () in the presence of long-range competing magnetic fields h(n) = h1/n3(n = 1,2,...,L) which are applied at opposite walls along the M-direction. Due to the fields, an interface between domains of different orientations that runs parallel to the walls forms and can be located close to one of the two surfaces or fluctuate in the centre of the film (localization–delocalization transition). This transition is the precursor of the wetting phase transition that occurs in the limit of infinite film thickness () at the critical curve Tw(h1). For T

When a magnetic Ising film is confined in a L × M geometry ( L ⪡ M ) short-range competing magnetic fields ( h 1 ) are applied at opposite walls along the M -direction, a (weakly rounded) localization–delocalization transition of the interface between domains of different orientation that runs parallel to walls can be observed. This transition is the precursor of a wetting phase transition that occurs in the limit of infinite film thickness ( L → ∞ ) at the critical curve T w ( h 1 ) . For T T w ( h 1 ) ( T > T w ( h 1 ) ) such an interface is bound to (unbound from) the walls, while right at T w ( h 1 ) the interface is freely fluctuating around the center of the film. We present extensive Monte Carlo simulations of Ising stripes in the L × M geometry, in order to describe both the localization–delocalization transition and the properties of the delocalized interface. To this aim, we take advantage of several available theoretical results. We make use of a suitable algorithm to define the local position of the interface along the film, such that its probability distribution can be used to account for the transition itself and the fluctuations in the local position of the interface (capillary waves). After describing the interface localization–delocalization transition, we pay attention to the properties of the delocalized interface with an emphasis on the effects of confinement. We analyze several quantities of interest in terms of the film thickness L . The width of the capillary waves ( s ) can be related to the width of the magnetization profiles ( w ) by means of a simple approximation. From this relation we estimate a value for the intrinsic width ( w 0 ) of the interface which agrees with the theoretical one. Also the correlation length ξ ∥ along the film is considered, and the behavior ξ ∥ ∼ L 2 compares very well to available exact results. Additionally, the interfacial stiffness β Γ obtained from the Fourier spectrum of the capillary waves reproduces the asymptotic theoretical value.

We study phase coexistence of an Ising ferromagnet in a bi-pyramid geometry with a square basal plane of linear extension 2L + 1. Antisymmetric surface fields act on the pyramid surfaces above and below the basal plane. In the limit L → ∞, the magnetisation stays zero at the bulk critical temperature, but becomes discontinuously non-zero at the cone filling critical temperature associated with a single pyramid. Monte Carlo simulations and scaling considerations show that this transition is described by a Landau theory with size-dependent coefficients that give rise to singular critical amplitudes.

The nature of the melting transition for a system of hard discs with translational degrees of freedom in two spatial dimensions has been analysed by a combination of computer simulation methods and a finite size scaling technique. The behaviour of the system is consistent with the predictions of the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory. The structural and elastic properties of binary colloidal mixtures in two and three spatial dimensions are discussed as well as those of colloidal systems with quenched point impurities. Hard and soft discs in external periodic (light-) fields show rich phase diagrams including freezing and melting transitions when the density of the system is varied. Monte Carlo simulations for detailed finite size scaling analysis of various thermodynamic quantities like the order parameter, its cumulants, etc, have been used in order to map the phase diagram of the system for various values of the density and the amplitude of the external potential. For hard discs we find clear indication of a reentrant liquid phase over a significant region of the parameter space. The simulations therefore show that the system of hard discs behaves in a fashion similar to charge stabilized colloids which are known to undergo an initial freezing, followed by a remelting transition as the amplitude of the imposed modulating field produced by crossed laser beams is steadily increased. Detailed analysis of the simulation data shows several features consistent with a recent dislocation unbinding theory of laser induced melting. The differences and similarities of systems with soft potentials (DLVO, 1/r12, 1/r6) and the relation to experimental data is analysed.

Confined magnetic Ising films in a L ? D geometry (), with short-range competing magnetic fields?(h) acting at opposite walls along the D-direction, exhibit a slightly rounded localization?delocalization transition of the interface between domains of different orientations that runs parallel to the walls. This transition is the precursor of a wetting transition that occurs in the limit of infinite film thickness () at the critical curve Tw(h). For T Tw(h)) such an interface is bounded (unbounded) to the walls, while right at Tw(h) the interface is freely fluctuating around the centre of the film. Starting from disordered configurations, corresponding to , we quench to the wetting critical temperature and study the dynamics of the approach to the stationary regime by means of extensive Monte Carlo simulations. It is found that for all layers parallel to the wall (rows), the row magnetizations exhibit a peak at a time and subsequently relax to the stationary, equilibrium behaviour. The characteristic time for such a relaxation scales as , as expected from theoretical arguments, that are discussed in detail.

A phenomenological theory of phase coexistence of finite systems near the coexistence curve that occurs in the thermodynamic limit is formulated for the generic case of d-dimensional ferromagnetic Ising lattices of linear dimension L with magnetization m slightly less than mcoex. It is argued that in the limit L→∞ an unconventional first-order transition occurs at a characteristic value mt

We present a Monte Carlo computer simulation and theoretical results for the dependence of the state diagram of a single semiflexible chain on the chain length. The calculated transition lines between different structures in the state diagrams for both studied chain lengths N=40 and N=80 can be described by theoretical predictions which include chain length dependence explicitly. The stability criteria of different structures are discussed. The theoretically predicted exponent in the dependence of the toroid size on the chain length is compatible with computer simulation results.

Square L ? L (L = 24?128) Ising lattices with nearest neighbour ferromagnetic exchange are considered using free boundary conditions at which boundary magnetic fields ? h are applied, i.e., at the two boundary rows ending at the lower left corner a field +h acts, while at the two boundary rows ending at the upper right corner a field ?h acts. For temperatures T less than the critical temperature Tc of the bulk, this boundary condition leads to the formation of two domains with opposite orientations of the magnetization direction, separated by an interface which for T larger than the filling transition temperature Tf (h) runs from the upper left corner to the lower right corner, while for T T > Tf (h) it scales as w ? L. The distribution P (?) of the interface position ? (measured along the z-direction from the corners) decays exponentially for T T > Tf (h). Furthermore, the Monte Carlo data are compatible with ? (Tf (h) ? T)?1 and a finite size scaling of the total magnetization according to M(L, T) = {(1 ? T/Tf (h))?? L} with ?? = 1. Unlike the findings for critical wetting in the thin film geometry of the Ising model, the Monte Carlo results for corner wetting are in very good agreement with the theoretical predictions.

We present a brief review of Monte Carlo simulations of ferromagnetic Ising lattices in a film geometry with surface magnetic fields. The seminal work of Nakanishi and Fisher [Phys. Rev. Lett. 49:1565 (1982)] showed how phase transitions in such models are related to wetting in systems with short range forces; and we will show how theoretical concepts about critical and tricritical wetting, interface localization-delocalization, and capillary condensation can be tested in this and similar models. After reviewing the qualitative, phenomenological description of these phenomena on a mean field level, we will summarize predictions of scaling theories. Comments will be made about the models studied and simulation techniques as well as the specific problems that occur in the relevant finite size scaling analysis. The resulting simulational data have prompted considerable new theoretical efforts, but there are still unsolved problems with respect to critical wetting. We will also present results for interface localization-delocalization transitions in both Ising models and lattice polymer mixtures in a thin film geometry and show that theory can account for many, but not all, aspects of the simulations. In systems with asymmetric boundary fields rather complex phase diagrams can result, and these should be relevant for corresponding experiments. The simulational evidence is fully compatible with the scaling predictions of Fisher and Nakanishi [J. Chem. Phys. 75:5875 (1981)] on capillary condensation. To conclude we shall summarize the major unanswered theoretical questions in this rich field of inquiry.

We report on a careful analysis of finite size effects on pressure measurements in Monte Carlo computer simulations of isotropic athermal solutions of flexible polymer chains by means of the repulsive wall method. We find finite size corrections to the pressure due to surface effects. These corrections are inversely proportional to the thickness of the simulation box, both if we keep the average density (in the canonical ensemble) or the chemical potential (in the grand canonical ensemble) constant in course of the preparation of the starting conformation. We propose a modification of the repulsive wall method which allows avoidance of these finite size effects and to estimate the pressure for an infinite system when running simulations in a finite box.

We use Monte Carlo simulations, single spin-flip as well as parallel tempering techniques to investigate the 10-state fully connected Potts glass for system sizes of up to N = 2560. We find that the α-relaxation shows a strong dependence on N and that for the system sizes considered the system remains ergodic even at temperatures below T D , the dynamical critical temperature for this model. However, if one uses the data for the finite size systems, such as the relaxation times or the time dependence of the spin autocorrelation function, and extrapolates them to the thermodynamic limit, one finds that they are indeed compatible with the results for N = ∞ (which are known from analytical calculations) if the extrapolation is done in the correct way. At low temperatures we find that relaxation times τ diverge like exp(c . N 1/2 ).

The dynamic and static properties of a supercooled (non-entangled) polymer melt are investigated via molecular-dynamics (MD) simulations. The system is confined between two completely smooth and purely repulsive walls. The wall-to-wall separation (film thickness), D, is varied from about 3 to about 14 times the bulk radius of gyration. Despite the geometric confinement, the supercooled films exhibit many qualitative features which were also observed in the bulk and could be analyzed in terms of mode-coupling theory (MCT). Examples are the two-step relaxation of the incoherent intermediate scattering function, the time-temperature superposition property of the late time alpha-process and the space-time factorization of the scattering function on the intermediate time scale of the MCT beta-process. An analysis of the temperature dependence of the alpha-relaxation time suggests that the critical temperature, T(c), of MCT decreases with D. If the confinement is not too strong ( D>or=10monomer diameter), the static structure factor of the film coincides with that of the bulk when compared for the same distance, T - T(c)( D), to the critical temperature. This suggests that T - T(c)( D) is an important temperature scale of our model both in the bulk and in the films.

We use Monte Carlo simulations to investigate the dynamic properties of the ten-state infinite-range Potts glass. By analyzing the spin autocorrelation function for system sizes up to N = 2560, we show that strong finite size effects are present around the predicted dynamic transition temperature. The autocorrelation function shows strong self-averaging at high temperatures, whereas close to the dynamic transition shows lack of self-averaging.

We consider the question of whether a two-dimensional hard-disc fluid has a first-order transition from the liquid state to the solid state as in the three-dimensional melting-crystallization transition or whether one has two subsequent continuous transitions, from the liquid to the hexatic phase and then to the solid phase, as proposed by Kosterlitz, Thouless, Halperin, Nelson and Young (KTHNY). Monte Carlo (MC) simulations of the fluid that study the growth of the bond orientational correlation length, and of the crystal are discussed. The emphasis is on a recent consistency test of the KTHNY renormalization group (RG) scenario, where MC simulations are used to estimate the bare elastic constants and dislocation fugacities in the solid, as a function of density, which then are used as starting values for the RG flow. This approach was validated earlier for the XY model as well.

We investigate by means of Monte Carlo simulations the fully connected p-state Potts model for different system sizes in order to see how the static and dynamic properties of a finite model compare with the, exactly known, behavior of the system in the thermodynamic limit. Using p=10 we are able to study the equilibrium dynamics for system sizes as large as N=2560. We find that the static quantities, such as the energy, the entropy, the spin glass susceptibility as well as the distribution of the order parameter P(q) show very strong finite size effects. From P(q) we calculate the forth order cumulant g_4(N,T) and the Guerra parameter G(N,T) and show that these quantities cannot be used to locate the static transition temperature for the system sizes investigated. Also the spin-autocorrelation function C(t) shows strong finite size effects in that it does not show a plateau even for temperatures around the dynamical critical temperature T_D. We show that the N-and T-dependence of the \alpha-relaxation time can be understood by means of a dynamical finite size scaling Ansatz. C(t) does not obey the time-temperature superposition principle for temperatures around T_D, but does so for significantly lower T. Finally we study the relaxation dynamics of the individual spins and show that their dependence on time depends strongly on the chosen spin, i.e. that the system is dynamically very heterogeneous, which explains the non-exponentiality of C(t).

Introduction In most of the discussion presented so far in this book, the quantum character of atoms and electrons has been ignored. The Ising spin models have been an exception, but since the Ising Hamiltonian is diagonal (in the absence of a transverse magnetic field), all energy eigenvalues are known and the Monte Carlo sampling can be carried out just as in the case of classical statistical mechanics. Furthermore, the physical properties are in accord with the third law of thermodynamics for Ising-type Hamiltonians (e.g. entropy S and specific heat vanish for temperature T → 0, etc.) in contrast to the other truly classical models dealt with in previous chapters (e.g. classical Heisenberg spin models, classical fluids and solids, etc.) which have many unphysical low temperature properties. A case in point is a classical solid for which the specific heat follows the Dulong–Petit law, C = 3 Nk B , as T → 0, and the entropy has unphysical behavior since S → –∞. Also, thermal expansion coefficients tend to non-vanishing constants for T → 0 while the third law implies that they must be zero. While the position and momentum of a particle can be specified precisely in classical mechanics, and hence the groundstate of a solid is a perfectly rigid crystal lattice (motionless particles localized at the lattice points), in reality the Heisenberg uncertainty principle forbids such a perfect rigid crystal, even at T → 0, due to zero point motions which ‘smear out’ the particles over some region around these lattice points. This delocalization of quantum-mechanical particles increases as the atomic mass is reduced; therefore, these quantum effects are most pronounced for light atoms like hydrogen in metals, or liquid helium. Spectacular phenomena like superfluidity are a consequence of the quantum nature of the particles and have no classical counterpart at all. Even for heavier atoms, which do not show superfluidity because the fluid–solid transition intervenes before a transition from normal fluid to superfluid could occur, there are genuine effects of quantum nature. Examples include the isotope effects (remember that in classical statistical mechanics the kinetic energy part of the Boltzmann factor cancels out from all averages, and thus in thermal equilibrium no property depends explicitly on the mass of the particles).

Dealing with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed-matter physics and statistical mechanics, this book provides an introduction to computer simulations in physics. This fourth edition contains extensive new material describing numerous powerful algorithms not covered in previous editions, in some cases representing new developments that have only recently appeared. Older methodologies whose impact was previously unclear or unappreciated are also introduced, in addition to many small revisions that bring the text and cited literature up to date. This edition also introduces the use of petascale computing facilities in the Monte Carlo arena. Throughout the book there are many applications, examples, recipes, case studies, and exercises to help the reader understand the material. It is ideal for graduate students and researchers, both in academia and industry, who want to learn techniques that have become a third tool of physical science, complementing experiment and analytical theory.

We present a study of the critical behavior of the Blume-Capel model with three spin states (S=±1,0) confined between parallel walls separated by a distance L where competitive surface magnetic fields act. By properly choosing the crystal field (D), which regulates the density of nonmagnetic species (S=0), such that those impurities are excluded from the bulk (where D=) except in the middle of the sample [where DM(L/2)≠], we are able to control the presence of a defect line in the middle of the sample and study its influence on the interface between domains of different spin orientations. So essentially we study an Ising model with a defect line but, unlike previous work where defect lines in Ising models were defined via weakened bonds, in the present case the defect line is due to mobile vacancies and hence involves additional entropy. In this way, by drawing phase diagrams, i.e., plots of the wetting critical temperature (Tw) versus the magnitude of the crystal field at the middle of the sample (DM), we observe curves of (first-) second-order wetting transitions for (small) high values of DM. Theses lines meet in tricritical wetting points, i.e., (Twtc,DMtc), which also depend on the magnitude of the surface magnetic fields. It is found that second-order wetting transitions satisfy the scaling theory for short-range interactions, while first-order ones do not exhibit hysteresis, provided that small samples are used, since fluctuations wash out hysteretic effects. Since hysteresis is observed in large samples, we performed extensive thermodynamic integrations in order to accurately locate the first-order transition points, and a rather good agreement is found by comparing such results with those obtained just by observing the jump of the order parameter in small samples. Fil: Trobo, Marta Liliana. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física de Líquidos y Sistemas Biológicos. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física de Líquidos y Sistemas Biológicos; Argentina Fil: Albano, Ezequiel Vicente. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física de Líquidos y Sistemas Biológicos. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física de Líquidos y Sistemas Biológicos; Argentina Fil: Binder, Kurt. Johannes Gutenberg Universitat Mainz; Alemania

The computation of interfacial free energies between coexisting phases (e.g.~saturated vapor and liquid) by computer simulation methods is still a challenging problem due to the difficulty of an atomistic identification of an interface, and due to interfacial fluctuations on all length scales. The approach to estimate the interfacial tension from the free energy excess of a system with interfaces relative to corresponding single-phase systems does not suffer from the first problem but still suffers from the latter. Considering $d$-dimensional systems with interfacial area $L^{d-1}$ and linear dimension $L_z$ in the direction perpendicular to the interface, it is argued that the interfacial fluctuations cause logarithmic finite-size effects of order $\ln (L) / L^{d-1}$ and order $\ln (L_z)/L ^{d-1}$, in addition to regular corrections (with leading order $\text{const}/L^{d-1}$). A phenomenological theory predicts that the prefactors of the logarithmic terms are universal (but depend on the applied boundary conditions and the considered statistical ensemble). The physical origin of these corrections are the translational entropy of the interface as a whole, "domain breathing" (coupling of interfacial fluctuations to the bulk order parameter fluctuations of the coexisting domains), and capillary waves. Using a new variant of the ensemble switch method, interfacial tensions are found from Monte Carlo simulations of $d=2$ and $d=3$ Ising models and a Lennard Jones fluid. The simulation results are fully consistent with the theoretical predictions., 27 pages, 16 figures

We present a unified scaling description for the dynamics of monomers of a semiflexible chain under good solvent condition in the free draining limit. We consider both the cases where the contour length $L$ is comparable to the persistence length $\ell_p$ and the case $L\gg \ell_p$. Our theory captures the early time monomer dynamics of a stiff chain characterized by $t^{3/4}$ dependence for the mean square displacement(MSD) of the monomers, but predicts a first crossover to the Rouse regime of $t^{2\nu/{1+2\nu}}$ for $\tau_1 \sim \ell_p^3$, and a second crossover to the purely diffusive dynamics for the entire chain at $\tau_2 \sim L^{5/2}$. We confirm the predictions of this scaling description by studying monomer dynamics of dilute solution of semi-flexible chains under good solvent conditions obtained from our Brownian dynamics (BD) simulation studies for a large choice of chain lengths with number of monomers per chain N = 16 - 2048 and persistence length $\ell_p = 1 - 500$ Lennard-Jones (LJ) units. These BD simulation results further confirm the absence of Gaussian regime for a 2d swollen chain from the slope of the plot of $\langle R_N^2 \rangle/2L \ell_p \sim L/\ell_p$ which around $L/\ell_p \sim 1$ changes suddenly from $\left(L/\ell_p \right) \rightarrow \left(L/\ell_p \right)^{0.5} $, also manifested in the power law decay for the bond autocorrelation function disproving the validity of the WLC in 2d. We further observe that the normalized transverse fluctuations of the semiflexible chains for different stiffness $\sqrt{\langle l_{\bot}^2\rangle}/L$ as a function of renormalized contour length $L/\ell_p$ collapse on the same master plot and exhibits power law scaling $\sqrt{\langle l_{\bot}^2\rangle}/L \sim (L/\ell_p)^\eta $ at extreme limits, where $\eta = 0.5$ for extremely stiff chains ($L/\ell_p \gg 1$), and $\eta = -0.25$ for fully flexible chains., Comment: 14 pages, 18 figures