Phase space compressibility

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

In general, it can be very difficult to determine the nature of the boundary terms. A specific result in an exactly solvable case is discussed in Section IV.A.2. Equation (55) is the Gallavotti-Cohen FT derived in the context of deterministic Anosov systems [28]. In that case, Sp stands for the so-called phase space compression factor. It has been experimentally tested by Ciliberto and co-workers in Rayleigh-Bemard convection [52] and turbulent flows [53]. Similar relations have also been tested in athermal systems, for example, in fluidized granular media [54] or the case of two-level systems in fluorescent diamond defects excited by light [55]. 

The quantity d/dV f is known as the phase space compressibility. For systems evolving according to Hamilton s equations of motion, the phase space compressibility vanishes ... 

Since, for Hamiltonian systems, the phase space compressibility vanishes, Eqs. [18] and [19] have the solutions (Fq F,) = (F Fq) = 1, from which it follows that the phase space volume element is the same at t = 0 as at an arbitrary time t ... 

The phase space compressibility corresponding to Eqs. [76] can be expressed as follows ... 

In this section we develop a general linear response theory for which the coupling of the system to the field in Eqs. [91] is small [i.e., fjt) 1]. Linear response theory should, in general, take into account the possibility of phase space compression coming from either the coupling to the external field or from the extended phase space variables. In traditional formulations of linear response theory, phase space compression has not been carefully considered. The present formulation treats the first level of compressibility exactly, for example, the compressibility due to the presence of an extended system (such as ther-... 

Boyd KJ, Lapicki A, Aizawa M, Anderson SL (1998) A phase-space-compressing, mass-selecting beamline for hyperthermal, focused ion beam deposition. Rev Sci Instrum 69 4106... 

The analysis from the previous section is used to write the metric tensor for this system, from the phase space compressibility, as... 

Assuming that this is the only conservation law for this system, then there exist no linear dependencies. Only 771 and the thermostat center r/c = however, are independently coupled to the dynamics. All other combinations of the thermostat variables k > 2 are trivial. As such, the phase space compressibility of the system can be written as... 

Following the same procedure that was outlined above, the phase space compressibility for this system is written as ... 

Trapping techniques have been and still are the basis of many new experiments in physics and chemistry. All these experiments make use of inherent advantages such as extremely long interaction times, the possibility to accumulate weak beams, phase space compression, laser cooling or interaction with buffer gas. This contribution has focused on the use of rf fields to explore collisions at low temperatures or with low relative velocities. The examples have shown that it is now possible to study collisions at energies of 1 meV or at temperatures of lOK. As already mentioned, there are activities to cool ions in traps to temperatures below 1 K using the slow tail of a cold effusive beam for buffer gas cooling. There are also efforts to heat ions with a laser in order to access temperatures above 2000 K. ... 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

The dissipative character of nonequilibrium systems is manifested in the compression of their phase space as referred to in Section 2.2. The phase-space compression factor is trivially related to the sum of the exponents... 

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