Loschmidt's paradox

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Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics and a time-symmetric formalism. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behaviour of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.

Johann Loschmidt's criticism was provoked by the H-theorem of Boltzmann, which was an attempt to explain using kinetic theory the increase of entropy in an ideal gas from a non-equilibrium state, when the molecules of the gas are allowed to collide. Loschmidt pointed out in 1876 that if there is a motion of a system from time t₀ to time t₁ to time t₂ that leads to a steady decrease of H (increase of entropy) with time, then there is another allowed state of motion of the system at t₁, found by reversing all the velocities, in which H must increase. This revealed that one of the key assumptions in Boltzmann's theorem was flawed, namely that of molecular chaos, that all the particle velocities were completely uncorrelated. One can assert that the correlations are uninteresting, and therefore decide to ignore them; but if one does so, one has changed the conceptual system, injecting an element of time-asymmetry by that very action.

Reversible laws of motion cannot explain why we experience our world to be in such a comparatively low state of entropy at the moment (compared to the equilibrium entropy of universal heat death); and to have been at even lower entropy in the past.

1 Arrow of time
2 Fluctuation theorem
3 The Big Bang
4 See also
5 References
6 External links

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Arrow of time

One possible resolution of the reversibility paradox is to hypothesize that there is a definite so-called arrow of time in the Universe. One possible mechanism for an arrow of time is to assume that time itself is defined by changes in cosmic entropy; another is to assume that low-level violations of time reversal symmetry at the particle physics level (namely the quantum arrow of time or, less probably, the weak arrow of time) are somehow driving cosmic entropy changes.

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Fluctuation theorem

One approach to handling Loschmidt's paradox is the fluctuation theorem, proved by Denis Evans and Debra J. Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain change in entropy over a certain amount of time. The theorem is proved with the exact time reversible dynamical equations of motion and the axiom of causality. The fluctuation theorem is proved utilizing the fact that dynamics is time reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments conducted by Eva M. Sevick-Muraca et. al. using optical tweezers apparatus. However, the fluctuation theorem assumes that the system is initially in an equilibrium state, so it can be argued that the theorem only verifies the time-asymmetry of the second law of thermodynamics based on an a priori assumption of time-asymmetric boundary conditions.

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The Big Bang

Another way of dealing with Loschmidt's paradox is to see the second law as an expression of a set of boundary conditions, in which our universe's time coordinate has a low-entropy endpoint: the Big Bang. From this point of view, the arrow of time is determined entirely by the direction that leads to the Big Bang, and a hypothetical universe with a maximum-entropy Big Bang would have no arrow of time. The theory of cosmic inflation tries to give reason why the early universe had such a low entropy.

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