Liouville equation entropies

Steeb, W. H. (1979). Generalized Liouville equation, entropy and dynamic systems containing limit cycles. Physica, 95A, 181-90. 

The evolution of an isolated system is then given by the classical and quantum Liouville equations for the fine-grained distribution functions (i.e., the evolution is entropy-preserving) ... 

At times t < f0 w [where f0 ° is an infinitesimal amount less than f0 ], the density is zero. Only after the pair is formed can there be any probability of its existence [499]. This is cause and effect, but strictly only applicable at a macroscopic level. On a microscopic scale, time reversal symmetry would allow us to investigate the behaviour of the pair at time and so it reflects the inappropriateness of the diffusion equation to truly microscopic phenomena. The irreversible nature of diffusion on a macroscopic scale results from the increase of entropy, and should be related to microscopic events described by the Sturm—Liouville equation (for instance) and appropriately averaged. 

With symmetric boundary conditions at the chosen time t = 0, the microscopic formulation conforms to time reversible laws as expected. The same conclusion follows from an analogous examination of the Liouville equation. In this setting, the initial data at time, t = 0, is a statistical density distribution or density matrix. Although there are celebrated discussions on the problem of the approach to equilibrium, we nevertheless observe that without course graining or any other simplifying approximations the exact subdynamics would submit to the same physical laws as above, i.e., time reversibility and therefore constant entropy. 

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. 

Before leaving this chapter, we briefly look at an important quantity known as Boltzmann s entropy, and we will examine reduced forms of the Liouville equation in generalized coordinates. 

In the next chapter, we will consider the nonequilibrium behavior of matter in the most general way by deriving the spatial and temporal variations in density, average velocity, internal and kinetic energy, and entropy. We will use the formal definitions of these quantities introduced in this chapter, including the possibility of their spatial and temporal variations via the probability density function described by the full Liouville equation. In the next chapter, we will also formally define local equilibrium behavior and look at some specific, well-known examples of such behavior in science and engineering. 

To preview the results somewhat, it will be shown that the general form of the transport equations contains expressions for the property flux variables (momentum flux P, energy flux q, and entropy flux s) involving integrals over lower-order density functions. In this form, the transport equations are referred to as general equations of change since virtually no assumptions are made in their derivation. In order to finally resolve the transport equations, expressions for specific lower-ordered distribution functions must be determined. These are, in turn, obtained from solutions to reduced forms of the Liouville equation, and this is where critical approximations are usually made. For example, the Euler and Navier-Stokes equations of motion derived in the next chapter have flux expressions based on certain approximate solutions to reduced forms of the Liouville equation. Let s first look, however, at the most general forms of the transport equations. 

Now, to obtain an entropy conservation equation, we can work with Eq. (5.9) modified to include the time dependence in a itself, or it is somewhat easier to work directly with the reduced Liouville equation, Eq. (3.20) or (3.24), for pairwise additive systems we choose the latter representation. 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

A second aspect of irreversibility concerns the approach to equilibrium. Since this question arises only for an isolated system, we will suppose the external forces to be turned off at some time, say < = 0, the system then being isolated for later times. Now the Navier-Stokes equations imply an approach to equilibrium, with monotonically increasing entropy. However, for times t > 0 the ensemble (51) is an exact solution to the Liouville equation for an isolated system, and as is well known the quantity S of Eq. (85) must then be time-independent and cannot approach its equilibrium value. The resolution of this paradox lies in the approximate nature of the Navier-Stokes equations, which hold only for slowly varying processes. The exact transport relations, taking into account processes of arbitrarily rapid variation, are evidently such as to maintain S constant. One may say that the low frequency contributions to S increase at the expense of the high frequency contributions, and measurements of a sufficiently coarse nature will show an apparent approach to equilibrium. Thus the approach to equilibrium is obtained as a natural consequence of an approximation method suited to slowly varying processes. 

The basic properties of the classical Liouville equation and the troublesome questions they raise are shared by the quantum Liouville equation. For the quantum case, we smmnarize these properties as follows (i) The canonical density operator is stationary with respect to the quantum Liouville equation, (ii) The quantum Liouville operator t is Hemtitiaa (iii) The quantum Liouville equation is time-reversal invariant, (iv) The Gibbs entropy S t) is time independent when S t) is determined using the formal solution p(t) of the quantum Liouville equatioa Given the density operator p(t), we can determine the average value d t)) of the quantum dynamical variable 0 at time t by using the relation... 

In Sections I.C and YD it was shown that the basic results from equilibrium and nonequilibrium thermodynamics can be established from statistical mechanics by starting from the maximum entropy principle. The success of this approach to the formulation of equilibrium and nonequilibrium thermodynamics suggests that the maximum entropy principle can also be used to formulate a general theory of nonequihbrium processes that automatically includes the thermodynamic description of nonequilibrium systems. In this section, we formulate a theory possessing this character by making use of a time-dependent projection operator P(t) that projects the thermodynamic description p,) of a system out of the global description pt) given by the solution of the Liouville equation. We shall refer to this theory as the maximitm entropy approach to nonequilibrium processes. 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

It might be possible to save the assumption in this context, by regarding it as an approximation. But this would introduce a major question of principle, since the motions which, without this approximate assumption are reversible, become, after it is made, irreversible One of the earliest applications Boltzmann made of his equation was to prove that the entropy integral increases, whereas Liouville s equation proves that it must remain strictly constant. In addition, the macroscopic determinacy appears to be established. This means that the assumption goes beyond the modest role of quantitative approximation, and assumes that of a new principle of physics—entering in a vague way through the back door. 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

Entropy production results by substituting the Liouville-von Neumann equation into this equation... 

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