Thermal bath system ensemble

This case is more rigorously treated in the theory of the Grand Canonical Ensemble , which consists of a number of identical systems that are able to exchange heat and particles with a common thermal bath. 

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 33], as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A Qiq) F( thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F( qk )- As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression... 

In contrast, a system in contact with a thermal bath (constant-temperature, constant-volume ensemble) can be in a state of all energies, from zero to arbitrary large energies however, the state probability is different. The distribution of the probabilities is obtained under the assumption that the system plus the bath constimte a closed system. The imposed temperature varies linearly from start-temp to end-temp. The main techniques used to keep the system at a given temperature are velocity rescaling. Nose, and Nos Hoover-based thermostats. In general, the Nose-Hoover-based thermostat is known to perform better than other temperature control schemes and produces accurate canonical distributions. The Nose-Hoover chain thermostat has been found to perform better than the single thermostat, since the former provides a more flexible and broader frequency domain for the thermostat [29]. The canonical ensemble is the appropriate choice when conformational searches of molecules are carried out in vacuum without periodic boundary conditions. 

The canonical ensemble corresponds to a system of fixed N and V, able to exchange energy with a thermal bath at temperature T, which represents the effects of the surroundings. The thermodynamic potential is the Helmholtz free energy, and it is related to the partition function Q ryT as follows ... 

The DFT-based CE Hamiltonian E(a) describes the energetics of the configuration space Xf (Ic) at T = 0 K. On condition that the physical system, where Xf (k) lives, is in contact with a thermal bath of temperature T, the ensemble of sites and their occupations are subject to heat flow and to thermal fluctuations of configuration, and each structure a is realized with a temperature-dependent probability p((r T). However, the simulation cells must have N 1000 sites for a realistic simulation of the temperature-dependent system behavior. This number of sites prevents the direct calculation of the partition function, of p((r T), and thus of the energy at thermal equilibrium,... 

The considerations above show that temperature is definable only for macroscopic (in the strict sense, infinitely large) systems, whose definition implies neglect of the disturbance during first thermal contact. The canonical ensemble is now seen to be characterized by its temperature and typically describes a system in equilibrium with a constant temperature bath. 

Molecular dynamics with periodic boundary conditions is presently the most widely used approach for studying the equilibrium and dynamic properties of pure bulk solvent,97 as well as solvated systems. However, periodic boundary conditions have their limitations. They introduce errors in the time development of equilibrium properties for times greater than that required for a sound wave to traverse the central cell. This is because the periodicity of information flow across the boundaries interferes with the time development of other processes. The velocity of sound through water at a density of 1 g/cm3 and 300 K is 15 A/ps for a cubic cell with a dimension of 45 A, the cycle time is only 3 ps and the time development of all properties beyond this time may be affected. Also, conventional periodic boundary methods are of less use for studies of chemical reactions involving enzyme and substrate molecules because there is no means for such a system to relax back to thermal equilibrium. This is not the case when alternative ensembles of the constant-temperature variety are employed. However, in these models it is not clear that the somewhat arbitrary coupling to a constant temperature heat bath does not influence the rate of reequilibration from a thermally perturbed... 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

In the canonical ensemble, the temperature of the system is controlled by its contact with a constant T reservoir of large heat capacity, generally called a heat bath . One way of including the thermal contact with the heat bath is to introduce an additional degree of freedom that represents the reservoir the simulation is then carried out for the extended system that includes the thermodynamic system of interest plus the reservoir. In the original implementation of this method (Nose, 1984a,b) thermal contact was achieved by scaling the particle velocities by a heat-bath variable s, so that... 

Physically, how could we obtain such an ensemble First, consider the laboratory temperature bath , sketched in Fig. 4.2. Experimentally, the system of interest is placed in the temperature bath where it can exchange energy with the bath. Now, if the bath is infinite in extent (infinite reservoir), the bath temperature remains constant. The so-called zeroth law of thermodynamics states that two systems in thermal contact with each other (i.e., energy flows freely between the two systems) will have the same temperature at equilibrium (i.e., at long contact times) thus, T = Tg if the contact time is sufficiently large. After a sufficiently long time, when the bath temperature and system temperature have equilibrated, the thermodynamic properties... 

This is the short-hand notation for a canonical ensemble, which represents the possible states of a statistical many particle system that is in thermal equilibrium with a heat bath in such an ensemble, particle number (N), volume (V) and temperature (T) are constant. 

The canonical ensemble represents a closed system at equilibrium with a macrostate specified by values of T, V, and N. The systems of the canonical ensemble are placed in thermal contact with each other so that each system of the ensemble is in a constant-temperature bath consisting of the other systems of the ensemble, and all have the same temperature. Figure 27.1 schematically depicts a portion of such an ensemble. 

Consider a system made up of a fixed number of particles N enclosed in a motionless container of fixed volume F. The temperature of the system is maintained at the temperature T by keeping the system in thermal contact with a large heat bath at the temperature T with which it is able to exchange energy. Such a system is called a closed, isothermal system. The macroscopic state of this system is called the canonical ensemble. 

From a thermodynamic point of view, molecular dynamics samples the microcanon-ical ensemble. In order to incorporate canonical thermal fluctuations by coupling the polymer system to a heat bath of canonical temperature T by means of a thermostat, Newton s equations of motion must be modified. Not the system energy E has to be constant, but the canonical expectation value ( ) at the given temperature. Thus, the task of the thermostat is twofold to keep the temperature constant and to sample the thermal... 

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