Time translation invariance

In physical systems it can happen that the transition probability densities are homogeneous in time and/or in space. A stochastic process X(t) is stationary if X(t) and X(t + r) obey the same probability laws for every r this means that all joint probability densities verify time translation invariance... 

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

When out-of-equilibrium dynamic variables are concerned, as will be the case in the following sections of this chapter, the equilibrium fluctuation-dissipation theorem is not applicable. In order to discuss properties such as the aging effects which manifest themselves by the loss of time translational invariance in... 

This general result simplifies considerably if the dynamics conserve a stationary distribution, Ps,(x). This condition is very general, applying to systems at equilibrium, nonequilibrium systems in a steady state, and nonequilibrium systems relaxing to equilibrium with time-translationally invariant dynamics. In this case, p and p are related in a simple way by microscopic reversibility,... 

There is clear that since the quantum fluctuation term does not depend on ending space coordinates but only on their time coordinates, so that in the end will depend only on the time difference since by means of energy conservation all the quantum fluctuation is a time-translation invariant, see for instance the Hamilton-Jacobi Eq. (4.41) therefore it may be further resumed as the fluctuation factor. 

To discuss the dynamics of a system near equilibrium one need only consider the equilibrium average of products of / s at different times. The average of itself contains no dynamical information, since due to time translational invariance,... 

Because the equilibrium ensemble is time translationally invariant, C is a function of t — t. Since we will be more interested in the Fourier and Laplace transforms of (19), we introduce the definitions... 

In the step from eqn [17] to eqn [18], we have explicitly used the time translation invariance property of the time-displaced correlation function. 

From the particle trajectories zjt(f), we evaluate the friction matrix M from Eq. (7.10). We make use of time-translational invariance to equivalently rewrite Eq. (7.10) as... 

We now turn to the evaluation of B. This is related to the zero eigenvalue solution as follows. Because of the time translation invariance (i.e., the invariance with respect to the time shift) the eigenvalue Equation (2.89) has a solution with zero eigenvalue (Xi = 0) as... 

Numerous properties of physical relevance can be deduced from the GFs. This becomes more evident if the GFs are Fourier transformed to energy space. Noticing that the one-particle GF and the p-p propagator are time-translational invariant, the transformation is readily performed ... 

The condition that the process a(t) is a stationary process is equivalent to the requirement tiiat all the distribution fimctions for a t) are invariant under time translations. This has as a consequence that W a, t) is independent of t and that 1 2(0, t 2, 2) depeirds on t = 2 -1. An even stationary process [4] has the additional requirement that its distribution fimctions are invariant under time reflection. For 1 2, this implies fV2(a 02> t) = 2 2 1 caWcd microscopic reversibility. It means that the quantities are even... 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

The first equation gives the diserete version of Newton s equation the second equation gives energy c onservation. We make two comments (1) Notice that while energy eouseivation is a natural consequence of Newton s equation in continuum mechanics, it becomes an independent property of the system in Lee s discrete mechanics (2) If time is treated as a conventional parameter and not as a dynamical variable, the discretized system is not tiine-translationally invariant and energy is not conserved. Making both and t , dynamical variables is therefore one way to sidestep this problem. 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

In linear response theory the optical activity is obtained from the part of the generalized susceptibility involving the temporal correlations of the electric and magnetic polarization fields46,47. For a system such as a normal fluid, described by a statistical operator that is invariant under space and time translations, the appropriate retarded Green function is,... 

Note that the stationary regime is obtained, not only by assuming that x(t, t ) is invariant by time translation (x(t, t ) = X (x)), but also by assuming that the age of the system tends toward infinity ... 

According to Noether s theorem (Arnold (1989)) symmetries of a mechanical system are always accompanied by constants of the motion. According to Section 3.1, system symmetries can be obvious (e.g. geometric) or hidden . Examples for obvious symmetries that lead to constants of the motion are invariance with respect to time translations, spatial translations and rotations. Invariance with respect to time leads to the conservation of energy, spatial and rotational symmetries lead to the conservation of linear and angular momentum, respectively (see, e.g., Landau and Lifschitz (1970)). Hidden symmetries cannot be associated with... 

As ahead stated, homogeneous, amorphous systems are assumed so that the stationary dishibution function is translationally invariant but anisotropic. The formal solution of the Smoluchowski equation for the time-dependent dishibution function... 

The time-dependent distribution function f (f) from (4a) can be used to show that a translationally invariant equilibrium distribution function leads to a trans-lationally invariant steady state distribution f, even though the SO in (3b) is not translationally invariant itself. To show this, a point in coordinate space (n,..., r y) shall be denoted by r, and shall be shifted, F —> F, with r, = r, -f a for all i a is an arbitrary constant vector. This gives... 

Translational invariance of sheared systems takes a special form for two-time correlation functions, because a shift of the point in coordinate space from F to F gives... 

---