Dissipative potential

Models for the dissipative dynamics can frequently be based on the assumption of fast decay of memory effects, due to the presence of many degrees of freedom in the s-region. This is the usual Markoff assumption of instantaneous dissipation. Two such models give the Lindblad form of dissipative rates, and rates from dissipative potentials. The Lindblad-type expression was originally derived using semigroup properties of time-evolution operators in dissipative systems. [45, 46] It has been rederived in a variety of ways and implemented in applications. [47, 48] It is given in our notation by... 

We have implemented this approach in studies of photodesorption, deriving a dissipative hamiltonian Fp — iWp/2, where the term Wp is a hermitian positive dissipative potential which gives relaxation rates kva = jh... 

Alternatively, the rate of instantaneous dissipation in the equation of the RDM is obtained from the dissipative potential form as... 

We introduce the vibrational wavefunctions VI (x) for each electronic state I and work with the vibronic basis set /, vj) and matrices in this basis, like V = [(J,vj Vp I,vi)] and W = [(vi Wu vj)]. The dissipative potential form of the instantaneous dissipative rate in the RDM equation follows from W as given above. 

Furthermore, in our previous work [29,30] we constructed a dissipative potential energy 2x2 matrix [W/j(x)[ resulting from p-electric-dipole/s-electric-... 

Results obtained from Eq.(22) and numerical solution of the RDM equation with the instantaneous dissipation from the dissipative potential formula, are in very good agreement with our previous results [29] using propagation of density amplitudes. The adsorbate state populations P/ reach at long times constant values, with Pg(oo) + Pe(oo) = 1 — Ps(oo) and Ps equal to the total population of the substrate, maintained by a steady interaction between p-and s-regions. 

Indeed, mh = 0 due to the skew symmetry of L (skew-symmetric multiplication of two identical terms gives always zero) and H mh =0 as a result of subtraction of two identical terms. The first line in (50) evaluated on J/l, v" is clearly identical to (55). The second line evaluated on is (55) multiplied by 0. The inequality sign is a consequence of the requirements put on the dissipation potential 2. 

The thermodynam ic force X that generates the dissipation will be assumed to be a standard friction force. This means that X = hp. For the sake of simplicity, we shall limit ourselves only to small forces and introduce the quadratic dissipation potential 5 = jAX2, where A>0 is a kinetic coefficient. [We can easily consider also non-quadratic potentials as, for example, S = A(exp X + exp(—X) — 2)]. 

The state variables are (41). The time evolution (63) does not involve any nondissipative part and consequently the operator L, in which the Hamiltonian kinematics of (41) is expressed, is absent (i.e., L = 0). Time evolution will be discussed in Section 3.1.3. We now continue to specify the dissipation potential 5. Following the classical nonequilibrium thermodynamics, we introduce first the so-called thermodynamic forces (X 1-.. X k) Jdriving the chemically reacting system to the chemical equilibrium. As argued in nonequilibrium thermodynamics, they are linear functions of (nj,..., nk,) (we recall that n = (p i = 1,2,..., k on the Gibbs-Legendre manifold) with the coefficients... 

The next problem is to find the dissipation potential that is a function of the thermodynamic forces, and Jp- is identical with the right-hand... 

The following question now arises What is the dissipation potential 2 that implies the mass action law fluxes In other words, we look for 5 that (i) satisfies (53), (ii) satisfies (67) with the fluxes given in (64), and (iii) in a small neighborhood of the equilibrium it reduces to (68). The answer is the following ... 

If we insert this dissipation potential into the right-hand side of (67) and use (66) then indeed we get the flux (64) with... 

We have just demonstrated that the multiscale nonequilibrium thermodynamics includes the mass action law of chemical kinetics as a particular case. The form (69) of the dissipation potential has been, at least implicitly, introduced already by Marcelin and de Donder (de Donder et al., 1936 Feinberg, 1972 Bykov et al., 1977 Gorban and Karlin, 2003, 2005). In the case when the thermodynamic potential does not have the specific form (66), the fluxes (67) are not exactly the same as the fluxes (64) given by the mass action law. Marcelin and de Donder have suggested that for the modified free energy the fluxes (67) should replace the fluxes given by the mass action law. 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

The fast time evolution, i.e., the time evolution describing the approach the slow manifold, is governed by (82) (i.e., M jw with the dissipation potential E given by (79) and with — J -). 

In the first section, the time rate of energy dissipation for a damped oscillator is introduced by the Rayleigh dissipation potential. The quantum version of this quantity, i.e., the time rate of the energy dissipation operator, can be originated from Equation (14) as... 

In the case of s linearly damped oscillator, the transformation of the Heisenberg picture into the Schrodinger picture by the method applied in classical quantum theory is impossible because the operator has a time-dependent part due to the dissipative process. Thus, a new way must be found to construct the wave equation of the oscillator. Kostin introduced a supplementary dissipation potential into his wave equation and constructed this dissipation potential by an assumption that the energy eigenvalues of the oscillator decay exponentially over time [39]. In Kostin s version of the wave equation, the operators are time independent, but the dissipation potential is nonlinear with respect to the wave function. In our theory, it is assumed that the abstract wave equation of the linearly damped oscillator has the form... 

We repeat the discussion of section 2.3.3 which culminated in eqn (2.46), but now clarified by the kinematic description of the previous section. Recall that the basic argument consisted of two parts. First, we construct a function (if the number of configurational coordinates is finite) or a functional (if the configurational coordinates are specified by a function) which is the sum of the time rate of change of the Gibbs free energy G( r, r, ) and a dissipative potential F( ri r ), of the form... 

The dissipative term in the L-vN equation has been derived in several ways. Expressions for the dissipative superoperator p (t) can be derived from the dynamics of the s-region, and could therefore be constructed from information about the electronic and atomic structure of this region. Three models of current interest are based on dissipative potentials [14,25], dissipative rate operators [26-29], and a second-order perturbation theory [4,6,30,31,40] with a coupling of p- and s-regions of the bilinear form = We develop the PWT treatment in the following, in terms applicable to some of these approaches. 

In comparison with the notebook, today s mobile phones have a much lower energy density. Using the same approach as above, we estimated the passive heat dissipation potential of a mobile phone housing with the following dimensions and boundary conditions ... 

Edelen, D.G.B. On the existence of symmetry relations and dissipation potentials. Arch. Ration. Mech. Anal. 51(3), 218-227 (1973)... 

D is the dissipative potential that can introduce viscous dampers in the equations of motions, Q contains the general forces, and Lt contains the Lagrange multipliers. 

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