The Physical Basis of The Direction of Time (The Frontiers ...

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66 3 The Thermodynamical Arrow of Time ∫ H gen [ρ(α)] := ρ(α)ln ρ(α) dα = ln p. (3.53) σ(α) The final form on the RHS is appropriate, since the mean probability p(α) for individual microscopic states and for given ρ(α) isthenp(α) =ρ(α)/σ(α). The entropy defined by −kH gen is also known as the relative entropy of ρ(α) with respect to the measure σ(α). The latter is often introduced ad hoc as part of a phenomenological description. Under the approximation w(α ′ ,α)=f(α)δ ′ (α − α ′ ) one now obtains the deterministic ‘drift’ limit of the master equation (3.51) – usually representing the first term of (3.44). It defines the first order of the Kramers–Moyal expansion for w(α, α ′ ), equivalent to an expansion of ρ(α ′ ,t) in terms of powers of α ′ −α at α ′ = α. The second order, w(α ′ ,α)=f(α)δ ′ (α−α ′ )+g(α)δ ′′ (α−α ′ ), leads to the Fokker–Planck equation as the lowest non-trivial approximation that leads to an irreversible equation (see de Groot and Mazur 1962, Röpke 1987). In this respect, it is analogous to the LAD equation as the lowest nontrivial order in the Taylor expansion of the Caldirola equation (2.31). A master equation is generally equivalent to a (stochastic) Langevin equation for individual macroscopic trajectories α(t) which may form a dynamical ensemble represented by ρ(α, t). In contrast to the Liouville equation (3.26), the master equation (3.48) or (3.36) cannot be unitary with respect to the inner product for probability distributions defined above (3.27). While total probability must be conserved by these equations, that of the individual trajectories cannot (see also Sect. 3.4). Information-reducing master equations describe an indeterministic evolution, which in general only determines an ever-increasing ensemble of different potential successors for each macroscopic state (such as a point in α-space). 7 As discussed above, this macroscopic indeterminism is compatible with microscopic determinism if that information which is transformed from relevant 7 The frequently used picture of a ‘fork’ in configuration space, characterizing a dynamical indeterminism, may be misleading, since it seems to imply unique predecessors. This would be wrong, as can be recognized, for example, in an equilibrium situation. In the case of a stochastic dynamical law that is defined on a finite set of states, a state must in general also have different possible predecessors, corresponding to an inverse fork. Inverse forks by themselves would represent a pure forward determinism (a ‘semigroup’, that may describe attractors). All these structures are meant to characterize the dynamical law. They are neither properties of the (f)actual history (which is assumed to evolve along a definite trajectory regardless of the nature of the dynamical law), nor of an evolving ensemble that represents a specific state of knowledge. However, only dynamically unique predecessors may give rise to recordable histories (consisting of ‘facts’ that are redundantly documented). The historical nature of our world is thus based on a uniquely determined or even overdetermined macroscopic past – see also footnote 1 of Chap. 2, Fig. 3.8, and Sect. 3.5. A macroscopic history that was completely determined from its macroscopic past would be in conflict with the notion of an (apparent) free will.
3.2 Zwanzig’s General Formalism of Master Equations 67 to irrelevant in the course of time no longer has any relevant (macroscopic) effects for all future times of interest. The validity of this assumption depends on the dynamics and on the specific initial conditions (3.46). Time-reversed (‘anti-causal’) effects could only be derived from an appropriate final condition by applying the corresponding approximations to (3.44) for tt 0 , but to non-increasing entropy for t
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the frontiers collection
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H. Dieter Zeh THE PHYSICAL BASIS OF
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Preface Four previous editions of t
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Contents Introduction .............
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Introduction The asymmetry of Natur
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Introduction 3 tion of time (thus a
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Introduction 5 1. Radiation. In mos
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Introduction 7 language is loaded w
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Introduction 9 between temporal and
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12 1 The Physical Concept of Time i
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14 1 The Physical Concept of Time r
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Page 48 and 49: 40 3 The Thermodynamical Arrow of T
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116 4 The Quantum Mechanical Arrow
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136 5 The Time Arrow of Spacetime G
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172 6 The Time Arrow in Quantum Cos
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Epilog Four weeks before his death,
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Epilog 201 The role of time is very
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Appendix: A Simple Numerical Toy Mo
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Notebook: A Toy Model Appendix: A S
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4. Two-Time Boundary Condition Appe
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References References marked with (
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References 211 Bläsi, B., and Hard
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References 213 Davies, P.C.W., and
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References 215 Gerlach, U.H. (1988)
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References 217 Horwich, P. (1987):
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References 219 Levine, H., Moniz, E
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References 221 Pearle, Ph. (1976):
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References 223 Smoluchowski, M. Rit
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References 225 Zeh, H.D. (1971): On
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Index absorber, 18, 24-28, 36-38, 5
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Index 229 forward determinism, 66,
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Index 231 stochastic process, see d
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