Quantum Thermodynamics

Now recall the simple but important example of averaging student grades as G 

We first consider the very large number of molecules as N as if they were distinguishable and then divide out the partial distinguishabdity due to the fact that M molecules are in a given energy level. In each of the i possible levels there are g, degenerate levels so the possible ways for M to be in one of a possible g,- levels is gf but when there are i levels, we end up with the factor H,- ( f ) the numerator. WeU, that is comphcated but we can think it through till we see that it is correct. 

Next we consider that S is entropy, which always tends to a maximum, so all we have to do is take the derivative of S and maximize it That is a real problem but we can just as well maximize 

what we really want to maximize is / = In ft - a — (B (M,Ej) where we maintain Nu3t = No + Ni + N2 = N-i- -----------------------------and also tot = NoEo + MiSi TNiSid-. That is, we want to 

The first step is easy and we obtain ----a Pe, = 0 where E, is the energy of the arbitrary fth 

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Enhanced sampling in conformational space is not only relevant to sampling classical degrees of freedom. An additional reason to illustrate this particular method is that the delocalization feature of the underlying distribution in Tsallis statistics is useful to accelerate convergence of calculations in quantum thermodynamics [34], We focus on a related method that enhances sampling for quantum free energies in Sect. 8.4.2. 

The quantum thermodynamic factor S is the quantum correction to the Kramers-Grote-Hynes classical result in the spatial diffusion limited regime, derived by Wolynes " ... 

A quantum thermodynamic definition of electronegativity has been provided by Gyftopoulos and Hatsopoulos [27] by considering the atom or the molecule as a member of a grand canonical ensemble where the energy (E) and the number of electrons (N) are continuous functions and all other properties of the ensemble are written in terms of these two independent variables. The chemical potential of the ensemble can be written as ... 

Gyftopoulos and Hatsopoulos [267] proposed a quantum thermodynamic definition of electronegativity (j) of a system where the electronic chemical potential (fi) is being considered as the negative of the same. 

From Eqs (3.37) and (3.38), it is obvious that for P = 1 one obtains the classical partition function and hence classical thermodynamics. The exact quantum partition function and quantum thermodynamics are obtained in the limit P —> oo however, in practice, it often suffices to take quite a small value of P to obtain accurate quantum results. 

In order to make the presented system of quantum thermodynamics more intelligible, we give three examples in an attempt to illustrate how some common observations within the frame of the above system should be interpreted. 

Even if it is not common yet in the current treatises on classical thermodynamics we can become aware that some alternative exposition of innovative structures of quantum thermodynamics [193,197] are plausible as build on the standard Carnot s theory where the fluxes of caloric are identified with the negative fluxes linked with information. For that reason, the thermal energy evolved by thermal processes becomes identical with the electromagnetic zero-point background energy evolved by the destruction of information inscribed in a structural unit ( qubit ). 

Proc. of 2 " IC on Frontiers of Mesoscopic and Quantum Thermodynamics, Prague 2004, (V. Spidka, P. Hubik, edts.), Phys. E., December 2005 (Elsevier, in print)... 

Now that we see that we can combine partition functions for all the quantized energy systems into a total partition function, we can think of other ways to use the quantized energy formulas. There is a curious history for this approach. We can see above that gvib is an important part of the total partition function and yet for many years low-resolution infrared spectra blurred many of the 3N — 6 vibrational modes of molecules typically larger than benzene. Thus the equations for quantum thermodynamics were known before 1940 but could only be applied to cases of small molecules in the gas phase using experimental vibrational frequencies. Since about 1985, quantum chemistry programs have included the calculation of vibrational frequencies with some correction factors that now make it possible to write down the full partition function by including theoretical... 

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