Second fluctuation-dissipation theorem

X. Because there is damping of nuclear motion into e-h pairs, excited e-h pairs at electron temperature Te must also be able to excite nuclear coordinates by fluctuating forces Fx that satisfy the second fluctuation dissipation theorem given as... 

Kubo calls this relationship the Second Fluctuation Dissipation Theorem. For its proof it should be noted that the modified Langevin equation can be written as... 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

This is the second fluctuation-dissipation theorem. It was shown that if y(t) is equal to C,Tb(t)IM, one recovers the original Langevin equation for v. 

Note that in the above expression the memory function is proportional to the auto-correlation function of the random force. This is the well-known second fluctuation-dissipation theorem. 

The simple friction coefficient f in Eq. [15a] is replaced in the generalized Langevin equation (Eq. [21]) by a friction kernel containing the memory function /(r). In addition, in place of the condition given by Eq. [16], the second fluctuation-dissipation theorem provides the more general condition (11)... 

F(t) is the random force, F(r) = eiQLlG, and G is iQLA. Equation (11.3.28) is the multidimensional second-fluctuation dissipation theorem. Again we note that F(t) is orthogonal to A, so that... 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

In equilibrium, or in a stable state, the magnitude of the fluctuations is the outcome of the competition between the jumps and the macroscopic return to equilibrium. Both effects are represented by the second and first term, respectively, on the right of (4.2b). This is the basis of the Einstein relation (VIII.3.9) and of the fluctuation-dissipation theorem. 

Remark. It is easily seen that the second term of (5.2) by itself causes the norm of if/ to change. In order that this is compensated by the fluctuating term the two terms must be linked, as is done by the relation U = V V. This resembles the classical fluctuation-dissipation theorem, which links both terms by the requirement that the fluctuations compensate the energy loss so as to establish the equilibrium. The difference is that the latter requirement involves the temperature T of the environment that makes it possible to suppress the fluctuations by taking T = 0 without losing the damping. This is the reason why in classical theory deterministic equations with damping exist, see XI.5. 

Equation (20) (or Eq. (21) in the classical limit) constitute the formulation of the fluctuation-dissipation theorem of the second kind or second FDT (using the Kubo terminology [30,31]). This theorem applies to the random force F(t), which is a bath dynamical variable. It expresses the fact that the bath is in equilibrium. 

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. 

Summing up, when the particle environment is a thermal bath, the two fluctuation-dissipation theorems are valid. In both theorems the bath temperature T plays an essential role. In its form (157) or (159) (Einstein relation), the first FDT involves the spectral density of a dynamical variable linked to the particle (namely its velocity), while, in its form (161) (Nyquist formula), the second FDT involves the spectral density of the random force, which is a dynamical variable of the bath. 

Another advantage of the simulation is its abihty to make direct tests on the range of validity of basic thermodynamical theorems such as the fluctuation-dissipation theorem. In the second paper of the series by Evans, he considers these points for the simplest type of torque mentioned above, —X F. Consider the return to equilibrium of a dynamical variable A after taking off at r = 0 the constant torque appUed prior to this instant in time. If the torque is removed instantaneously, the first fluctuation-dissipation theorem implies that the normalized fall transient will decay with the same dependence as the autocorrelation function (A(t)A(O))- Al /(A 0)) — Therefore,... 

In this paper we gave a dynamic extension of the DFT, by deriving a L-D equation (11) with the fluctuation-dissipation theorem (9). We showed that the stochastic equation correctly samples the density field according to the probability exp —jflf [n], (17), based on the second H-theorem (16). At this point we note however that our TO-DFT is phenomenological md it is desirable to have a first-principle dynamics generalization of DFT. 

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

Appendix B The Second Law from the Fluctuation-Dissipation Theorem... 

APPENDIX B THE SECOND LAW FROM THE FLUCTUATION-DISSIPATION THEOREM... 

In Appendix B, we will demonstrate the second law from the fluctuation-dissipation theorem. 

Coulomb contributed what is often called the third law of friction, i.e. that is relatively independent of sliding velocity. The experiments discussed in Section I.D show that the actual dependence is logarithmic in many experimental systems and that often increases with decreasing velocity. Thus there is a fundamental difference between kinetic friction and viscous or drag forces that decrease to zero linearly with v. A nearly constant kinetic friction implies that motion does not become adiabatic even as the center-of-mass velocity decreases to zero, and the system is never in the linear response regime described by the fluctuation dissipation theorem. Why and how this behavior occurs is closely related to the second issue raised above. 

Here the first two terms just give ma = Force as mass times the second time derivative of the friction equal to the F as the negative derivative of potential, y is the memory friction, and F(t) is the random force. Thus the complex dynamics of all degrees of freedom other than the reaction coordinate are included in a statistical treatment, and the reaction coordinate plus environment are modeled as a modified one-dimensional system. What allows realistic simulation of complex systems is that the statistics of the environment can in fact be calculated from a formal prescription. This prescription is given by the Fluctuation-Dissipation theorem, which yields the relation between the friction and the random force. In particular, this theory shows how to calculate the memory friction from a relatively short-time classical simulation of the reaction coordinate. The Quantum Kramers approach. 

The first-principle method is being developed for systems with long-range dispersion forces. There are two ways to include dispersion forces in first-principle calculations. A semiempirical van der Waals interaction can be taken into account in ab initio calculations. It is realized by using the Lenard-Jones potential of the form (11.26). The second approach is based on the adiabatic connection fluctuation-dissipation theorem. This theory includes seamless long-range dispersion forces... 

The first terra on the right side comes from a kind of surface tension and tends to smooth the surface, while the second term is a Gaussian fluctuating white noise satisfying the fluctuation dissipation theorem. Equation (31.6) leads for it = 2 to Ds = 1-5. Also numerically, this result is well verified for random deposition with surface diffusion [34—38]. For d = 3, we find from Eq. (31.6) = 3. In... 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

The positivity of spectral matrix Cab u ) leads to the positivity of the Hermite matrix Jo6(o ) for a > 0. The relation between Jabioj) and Cab < ) as shown in the second identity of Eq. (2.9) is called the generalized fluctuation-dissipation theorem, which can be equivalently expressed... 

FIGURE 1 Distribution of work values for many repetitions (realizations) of a thermodynamic process involving a nanoscale system. The process might involve the stretching of a single RNA molecule or, perhaps, the compression of a tiny quantity of gas by a microscopic piston. The tail to the left of AF represents apparent violations of the second law of thermodynamics. The distribution p( W) satisfies the nonequilibrium work theorem (Eq.6), which reduces to the fluctuation-dissipation theorem (Eq.l) in the special case of near-equUibrium processes. 

The second, more practical, expression is in terms of the dynamic magnetic susceptibility x iQ> T). Its connection to the correlation function described by eq. (2) is given via the fluctuation dissipation theorem,... 

Equating the integrants on both sides yields a second form of the fluctuation-dissipation theorem... 

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