Thermal bath system

It is known that the interaction of the reactants with the medium plays an important role in the processes occurring in the condensed phase. This interaction may be separated into two parts (1) the interaction with the degrees of freedom of the medium which, together with the intramolecular degrees of freedom, represent the reactive modes of the system, and (2) the interaction between the reactive and nonreactive modes. The latter play the role of the thermal bath. The interaction with the thermal bath leads to the relaxation of the energy in the reaction system. Furthermore, as a result of this interaction, the motion along the reactive modes is a complicated function of time and, on average, has stochastic character. 

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

We will find the probability P(t) for the system to pass the point q = q0/2 up to the moment of time t. This probability gives the upper estimate for the transition probability since, in principle, there are trajectories for which the system goes back to the left potential well after crossing the top of the potential barrier. However, if the contribution of these trajectories is small, as is the case for not too strong an interaction with the thermal bath at large narrow barriers, P(t) is close to the exact value of the transition probability. 

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 two ends of the system are put into contact with thermal baths at temperature Tl and Tr for left and right bath, respectively. In fact, Eq. (5) is the Hamiltonian of the Frenkel-Kontorova (FK) model which is known to have normal heat conduction(Hu Li Zhao, 1998). For simplicity we set the mass of the particles and the lattice constant m = a = 1. Thus the adjustable parameters are (ki, hnt, kR, Vl, Vr, Tr, Tr), where the letter L/R indicates the left/right segment. In order to reduce the number of adjustable parameters, we set Vr = XVr, kR = Xki, Tl = T0(l + A),Tr = To(l — A) and, unless otherwise stated, we fix Vl = 5, ki = 1 so that the adjustable parameters are reduced to four, (A, A, kint, To)- Notice that when A > 0, the left bath is at higher temperature and vice versa when A < 0. 

This equation has been used by Sundstrom and coworkers [151] and adapted to the analysis of femtosecond spectral evolution as monitored by the bond-twisting events in barrierless isomerization in solution. The theoretical derivation of Aberg et al. establishes a link between the Smoluchowski equation with a sink and the Schrodinger equation of a solute coupled to a thermal bath. The reader is referred to this important work for further theoretical details and a thorough description of the experimental set up. It is sufficient to say here that the classical link is established via the Hamilton-Jacobi equation formalism. By using the standard ansatz Xn(X,t)= A(X,i)cxp(S(X,t)/i1l), where S(X,t) is the action of the dynamical system, and neglecting terms in once this... 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

This introduces a new unknown (and free at this stage) coefficient K. In the case of a system of molecules in a thermal bath (definitely not the one we consider), there is a relation between D and K such that, at thermal equilibrium, the equilibrium density in the potential (h is given by Boltzmann s law. This requires that K = mD/hgT, where kg is Boltzmann s constant and T the absolute temperature. In Eq. (12) the factor p in front of in is to ensure that, if... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

The system is assumed to be in contact with a thermal bath at temperature T. We also assume that the microscopic dynamics of the system is of the Markovian type the probability that the system has a given configuration at a given time only depends on its previous configuration. We then introduce the transition probability Wt(C C ). This denotes the probability for the system to change from C to C at time step k. According to the Bayes formula,... 

Quantum-state decay to a continuum or changes in its population via coupling to a thermal bath is known as amplitude noise (AN). It characterizes decoherence processes in many quantum systems, for example, spontaneous emission of photons by excited atoms [35], vibrational and collisional relaxation of trapped ions [36] and the relaxation of current-biased Josephson junctions [37], Another source of decoherence in the same systems is proper dephasing or phase noise (PN) [38], which does not affect the populations of quantum states but randomizes their energies or phases. 

We focus on two regimes a two-level system coupled to either an AN or PN thermal bath (Figure 4.5). The bath Hamiltonian (in either regime) will be... 

We first consider the AN regime of a two-level system coupled to a thermal bath. We will use off-resonant dynamic modulations, resulting in AC-Stark shifts (Figure 4.5(a)). The Hamiltonians then assume the following form ... 

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... 

Assume that a noninteracting nanosystem is coupled weakly to a thermal bath (in addition to the leads). The effect of the thermal bath is to break phase coherence of the electron inside the system during some time Tph, called decoherence or phase-breaking time. rph is an important time-scale in the theory, it should be compared with the so-called tunneling time - the characteristic time for the electron to go from the nanosystem to the lead, which can be estimated as an inverse level-width function / 1. So that the criteria of sequential tunneling is... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

Thus, even in the presence of pumping, of energy Sk, from the thermal bath into the modes k, a Planck-type distribution of photons over the modes k, and hence a Boltzmann-type distribution of molecular systems over vibrational states (v, V2- -vj,..vk,..vz) results from linear systems. 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

A system well-adapted to the analysis of these concepts is a diffusing particle in contact with an environment, which itself may be in equilibrium (thermal bath) or out of equilibrium (aging medium). 

Let us assume that, at time t = f,, the density operator of the global particle-plus-bath system is factorized in the form ppart where pbath denotes the thermal equilibrium density operator of the unperturbed bath and ppart, the particle density operator. 

One usually studies diffusion in a thermal bath by writing two fluctuation-dissipation theorems, generally referred to as the first and second FDTs (using the Kubo terminology [30,31]). As recalled for instance in Ref. 57, the first FDT expresses a necessary condition for a thermometer in contact solely with the system to register the temperature of the bath. As for the second FDT, it expresses the fact that the bath itself is in equilibrium. 

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