Extended ensemble

Order Parameters, Reaction Coordinates, and Extended Ensembles... 

It is possible to treat the parameter A in the alchemical transformation as a dynamic variable using an extended ensemble [56]. For simplicity of implementation, it has been proposed to use two variables A0 and A i such that A +A = 1. The Hamiltonian function is then defined as [57, 58]... 

The tunneling problem at first order phase transitions and for disordered systems, where tunneling times often diverge exponentially can be overcome using extended ensemble methods, which are the main topic of the next chapter. 

An alternative to these extended ensembles for the simulation of frustrated magnets is the parallel tempering or replica exchange Monte Carlo method [22-25], Instead of performing a single simulation at a fixed temperature, simulations are performed for M replicas at a set of temperatures Ti,T2,. .., Tm- In addition to standard Monte Carlo updates at a fixed temperature, exchange moves are proposed to swap two replicas between adjacent temperatures. These swaps are accepted with a probability... 

Analog to the argument for the extended ensemble in energy space the current j is maximized by choosing a probability distribution... 

Figure 5 illustrates the optimized temperature sets for the Ising ferromag-net obtained by several iterations of the above feedback loop. After feedback of the local diffusivity temperature points accumulate near the critical temperature Tc = 2.269 of the transition. This is in analogy to the optimized histograms for the extended ensemble simulations where resources where shifted towards the critical energy of the transition, see Fig. 2. 

Extended ensemble methods, such as the multicanonical ensemble, Wang-Landau sampling or parallel tempering can also be generalized to quantum systems [35,36], as we will show in the next two sections. 

Thermodynamic quantities that cannot be expressed as the expectation value of an operator, such as the free energy or entropy cannot be directly measured but require an extended ensemble simulation, discussed in Sect. 5. 

In this section we will present generalizations of extended ensemble simulations to world line quantum Monte Carlo simulations, in particular ... 

Another advantage of extended ensemble simulations is the ability to directly calculate the density of states and from it thermodynamic properties such as the entropy or the free energy that are not directly accessible in canonical simulations. In the following we will again use quantum magnets as concrete examples. A generalization to bosonic and fermionic models will always be straightforward. 

Mullinax JW, Noid WG (2009) Extended ensemble approach for deriving transferable coarse-grained potentials. J ChemPhys 131 104110... 

An approach for tracking electronic degrees of freedom in parallel with a numerical integration of the classical equations of motion for the nuclei, and therefore determining V(r ) on the fly, has been devised by Car and Parrinello [27]. This extended ensemble molecular dynamics method, termed ab initio molecular dynamics, solves the electronic problem approximately using the Kohn Sham formulation of Density Functional Theory. This approach proved useful for covalent systems it still has to be applied to the systems where the properties of interest are defined by Lennard-Jones interactions. 

To further illustrate how extended ensembles can be designed to conduct MD simulations under various macroscopic constraints, we discuss here the NTLxPyycr ensemble. NTL yyOzz is an appropriate statistical ensemble for the simulation of uniaxial tension experiments on solid polymers [12] or relaxation experiments in uniaxially oriented polymer melts [13]. This ensemble is illustrated in Fig. 2. The quantities that are kept constant during a molecular simulation in this ensemble are the following ... 

In PT, not a single system but an extended ensemble of n systems, labeled... 

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