Time translation

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... 

Each bin is connected to a memory location in a computer so that each event can be stored additively over a period of time. All the totaled events are used to produce a histogram, which records ion event times versus the number of times any one event occurs (Figure 31.5).With a sufficiently large number of events, these histograms can be rounded to give peaks, representing ion m/z values (from the arrival times) and ion abundances (from the number of events). As noted above, for TOP instruments, ion arrival times translate into m/z values, and, therefore, the time and abundance chart becomes mathematically an m/z and abundance chart viz., a normal mass spectrum is produced. 

Time Translation of Cost Data. Cost data for any particular point in time can be corrected to the present or any other time by means of cost indexes in the relation ... 

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

Let us, therefore, assume that the amplitude >ft x) describing a relativistic spin particle is an -component object. We are then looking for a hermitian operator H, the hamiltonian or energy operator, which is. linear in p and has the property that H2 = c2p2 + m2c4 = — 2c2V2 + m2c4. We also require H to be the infinitesimal operator for time translations, i.e., that... 

Furthermore, since H, the hamiltonian, is the time translation operator... 

The existence of a unitary transformation U(a,A) which relates the field operators in the two frames imposes certain conditions on the operators themselves they must satisfy certain commutation rules with P and Muv. Consider first the case of a space-time translation... 

For n 0 the above equation must coincide with the equation of motion, hence P0 is the hamiltonian of the system. Similarly one deduoes that the P, (i = 1,2,3) must be the components of the total momentum operator. We call the operator PB the displacement operators for spaoe time translations in the sense that for an arbitrary operator F(x) which is a function of tfi(x) and A (x)... 

Observe that eq. (3) engenders a tme change so long H(x) is the generator of time translations i.e. a change in the direction cosines can be experimentally detectable as for instance in a pump-probe experiment or in electron photodetachments measurements. 

Proust, Marcel, In Search of Lost Time (translated by C. K. Moncrieff and Terence Kilmartin revised by D. J. Enright) (New York Modem Library, 1992, 6 volumes). 

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 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... 

The symmetry of the problem can be used to analyze the structure of such a second-order contribution. The spin-rotational symmetry (about the B-field s direction) and the time-translational symmetry imply that (i) the longitudinal and transverse fluctuations, Xj and X, do not interfere and may be considered separately (ii) it is convenient to expand the transverse fluctuating field in circularly polarized harmonic modes, and the latter contribute independently. 

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)]. 

The coke profiles in the reactor bed can be predicted excellently by the model as shown by the solid lines in Figure 1. Figure 2 shows good consistency is also obtained for the average coke content over the reactor bed versus time on stream. Note that within the time period of reactor startup plus one hour of operation, the average coke content of the reactor bed is already at about 5 wt%. The model cannot be applied to this startup and initial period with the rapid transients of temperature, activity "spike" and concentration. However, compensation for this interval can be made by a time translation of the model a model time of 36 hours is fixed at an experimental time of zero. A temperature difference of more than 20C between the center of the bed and outer wall of the reactor in the startup stage has been observed in our laboratory for some experiments. About three-fourths of this difference is across the catalyst bed itself. Startup of the reactor at reasonably lower temperatures in order to control the coke formation and to better maintain the catalyst activity is important, if not critical. 

Modern branded products must be unvarying in their quality. At each new purchase the consumer expects to find a product that is indistinguishable from the one bought the last time. Translated into a specification for perfumes, this means that the perfume must be stable in the product or, to put it more precisely, it must undergo no noticeable change, not in odor character nor in intensity nor in color, during the entire shelf life of the product. 

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,... 

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... 

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... 

In the MD simulations, one makes use of the time translational symmetry for a stationary process and average the TCP over time, t. 

Theorem 22.2 (Time-Translation Independence) The output depends only on the input. This means that, if the input signal is advanced or delayed by a time increment, the output will be advanced or delayed by the same time increment. Thus, x(t + r) corresponds to f(t + r) such that g t,t ) can depend only on the difference between t and t. Equation (22.1) can be written... 

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