Canonical temperature

Case Study on M7 in Terms of the Single Exponential Form Appendix B On Ergodicity and Nonergodicity of the Liquid-like Dynamics Appendix C Canonical Temperature... 

Latora et al. [18] discussed a relation between the process of relaxation to equilibrium and anomalous diffusion in the HMF model by comparing the time series of the temperature and of the mean-squared displacement of the phases of the rotators. They showed that anomalous diffusion changes to a normal diffusion after a crossover time, and they also showed that the crossover time coincides with the time when the canonical temperature is reached. They also claim that anomalous diffusion occurs in the quasi-stationary states. 

To observe the relaxation process, we use the magnetization M(t). Note that observing M(t) corresponds to observing 2K t) / N by using Eq. (3), and 2K(t) /TV is the time series of the temperature, since the canonical average of 2K/N coincides with the canonical temperature. 

The macroscopic variables U, S, V, and N are all extensive variables. Ratios of extensive variables are intensive variables and thus do not linearly depend on the system size. In analogy to Eq, (2,3), it is therefore suitable to express the intensive variable canonical temperature by the derivative of U with respect to S,... 

The canonical temperature T of the closed system is identical with the microcanonical heat bath temperature, i.e., T = However, this does not mean that also the canonical and microcanonical system temperatures coincide actually = Tis only valid in the thermodynamic limit, where the energetic fluctuations... 

Finally, after the best possible estimate for the multicanonical weight function is obtained, a long multicanonical production ran is performed, including all measurements of quantities of interest. From the multicanonical trajectory, the estimate of the canonical expectation value of a quantity O is then obtained at any (canonical) temperature Thy. 

Since multicanonical sampling effectively works at infinite canonical temperature, we use (4,153) to express the partition sum that then coincides with the total number of all possible conformations as... 

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... 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

But what if surface fluctuations are non-negligible In this case, the canonical temperature can be a badly defined control parameter for studies of nucleation transitions with phase separation. This becomes apparent in the following microcanonical folding analysis of the hydrophobic-polar heteropolymer sequence 20.6, whose canonical thermodynamic and kinetic properties have been investigated in detail in the previous section, by employing the AB model. 

It is sometimes argued that proteins fold in solvent, where the solvent serves as a heat bath. This would provide a fixed canonical temperature such that the canonical interpretation is sufficient to imderstand the transition. However, the solvent-protein interaction is actually implicitly contained in the heteropolymer model and, nonetheless, the microcanonical analysis reveals this effect which is simply lost by integrating out the energetic fluctuations in the canonical ensemble (see Fig. 9.14). 

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