Statistical mechanics ensembles

Monte Carlo Methods. Although several statistical mechanical ensembles may be studied using MC methods (2,12,14), the canonical ensemble has been the most frequently used ensemble for studies of interfacial systems. In the canonical ensemble, the number of molecules (N), cell volume (V) and temperature (T) are fixed. Hence, the canonical ensemble is denoted by the symbols NVT. The choice of ensemble determines which thermodynamic properties can be computed. 

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

De Pablo, Laso, and Suter studied the behavior of polyethylene above and below the melting point." A variation of the MC technique was used to study polyethylene and to sample its configuration space. The technique is suitable for the study of long chains at high densities. The simulations were carried out in an isobaric—isothermal statistical mechanical ensemble, which allows the calculation of density at a given pressure and temperature. A series of simulations at different temperatures indicated a phase transition. The polyethylene model employed in the simulations crystallizes spontaneously at low temperatures. At temperature higher than the melting point, the simulated melt is described accurately by the model. 

Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system s free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is ... 

B.C. Eu, Non-Equilibrium Statistical Mechanics, Ensemble Method, Academic Kluwer Publishers, Dordrecht, 1998. 

In order to solve Newton s equations of motion they are discretised in time. Various schemes exist, but for the purpose of this discussion, it makes little difference which one is used. We employ the leap-frog scheme [33]. Newton s equations of motion conserve the total energy of the system and lead to a micro-canonical statistical-mechanical ensemble. In practice, one is usually more interested in ensembles in which the temperature (canonical ensemble) or the temperature and the pressure (isothermal-isobaric ensemble) are conserred. For this, Newton s equations of motion have to be slightly modified to couple the system temperature T (or pressure p) to a temperature (or pressure) bath of temperature T (or pressure po). There are several such constant-temperature and constant-pressure schemes [33, 39], We use the loose-coupling algorithm [40] which implements a first-order thermostat and manostat. 

The equations of motions can be formulated in the Newtonian, Hamiltonian, or Lagrangian approaches, all of which correspond to the microcanonical (NVE) statistical-mechanical ensemble in which the number of atoms, N, volume, V, and total energy, E, are conserved. Simulations in other statistical-mechanical ensembles can... 

Simulations generally involve a fixed number of particles (N). To model bulk systems with a finite number of particles, periodic boundary conditions are applied. The volume (V) of the periodically repeating simulation box may be kept fixed or varied. Extensions to the scheme in Eq. (2) allow coupling the system to a thermostat and/or barostat to control the temperature T and pressure P of the system [22-24]. Hence, depending on the chosen conditions, the distribution sampled during the MD simulation is representative of either the microcanonical (NVE), canonical (NVT), or isothermal-isobaric (NPT) statistical mechanics ensembles [19]. 

MD is readily applicable to a wide range of models, with and without constraints. It has been extended from the original microcanonical ensemble formulation to a variety of statistical mechanical ensembles. It is flexible and valuable for extracting dynamical information. The Achilles heel of MD is its high demand of computer time, as a result of which the longest times that can be simulated with MD fall short of the longest relaxation times of most real-life... 

Here H(p, q) is the Hamiltonian (the sum of the potential and kinetic energy) of the system and p,- are the generalized momenta conjugate to qt. If the generalized coordinates represent the atomic coordinates of a molecular system, this approach is referred to as molecular dynamics. If one makes the assumption that the resulting trajectories cover phase space (or more specifically, are ergodic) then they generate a statistical mechanical ensemble. ... 

Several approaches for calculating the surface tension have been developed and are briefly summarized here. The fundamental definition of the surface tension y depends on the statistical mechanical ensemble used. For example, at constant N, V, T ... 

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