The dynamical fundamental equations (Newton, Lagrangian and Hamiltonian mechanics, Maxwell's equations, Schrodinger equation) are the energy conservation equations, and correspond to the first law of thermodynamics. Time reversal invariant is the basic symmetry of these equations, it is usually considered as a criterion for reversibility: T-symmetry means reversible.
However, the first law of thermodynamics is also time reversal invariant, and irreversibility is the conclusion of the second law, not that of the first law, such that T-symmetry of the first law cannot be a criterion for irreversibility of the second law.
Similar to that we cannot derive irreversibility from the first law of thermodynamics, in my opinion, “the fundamental structure of theoretical physics might be divided into the two main lines according to different laws, one being the first law physics, the other one being the second law physics”[1]. The conservation equations (CPT-symmetry) correspond to the first law physics, and irreversibility corresponds to the second law physics. T-symmetry of the conservation equations (the first law physics) cannot be considered as a criterion for irreversibility of the second law (physics). It is the symmetry of the equations, we are not sure if it is also the symmetry of the dynamic phenomena themselves.
Then how do we prove the conclusion that the fundamental dynamic processes are reversible?
[1] https://www.researchgate.net/publication/236983852_arXiv1201.4284v4
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Dear Charles,
The question related to theoretical criterion, and some phenomena can hardly be confirmed by examination of the law. For example, Maxwell's equations are time reversal invariant, but the mirror images of time reversal of radiant electromagnetic fields have never been found. In this example, we can noly confirm that T-symmetry is the symmetry of Maxwell's equations, but can not confirm that it is also the symmetry of radiant electromagnetic fields themselves. The transmission of radiant electromagnetic fields is an unidirectional process.
Dear Robert and Charles,
1) Maxwell's equations are determinism and have no any probability factor, so there has no “the overwhelming probability”.
2) A broken glass cannot reassemble and jump back up onto the table, the reason lies in mechanical equilibrium, not in “the overwhelming probability”. Please note that the postulate of the equal a priori probability does not hold here, and mechanical equilibrium is one of thermodynamic equilibrium conditions.
3) Low entropy is only the initial conditions, not the mechanism or principle of irreversibility. The mechanism or principle of irreversibility explains why entropy increases.
Please note that the postulate of the equal a priori probability does not always hold for generalized thermodynamic system. The world we seen is not a gas in box.
Dear Robert and Charles,
Sorry and Thanks. I misunderstood the meaning that the “overwhelming probability” denotes “a given initial condition”, and I partly agree with the explanation on the overwhelming probability of a given initial condition. “partly” means that there are some exceptions.
But the “overwhelming probability” of “a given initial condition” does not seem to have explained the question. Let me rephrase my question as below.
1) Time reversal invariant is the basic symmetry of dynamic laws, it is usually considered as a criterion for reversibility: T-symmetry means reversible.
2) The first law of thermodynamics is also time reversal invariant, but thermodynamic processes can be irreversible, so T-symmetry of the first law cannot be a reversibility criterion of thermodynamic processes. In this case, time reversal invariant is only the symmetry of the law or equation, not that of the phenomena themselves.
3) Compare 1) with 2), we can noly confirm that time reversal invariant is the symmetry of the dynamic laws or the equations. How do we prove that T-symmetry of dynamic laws can be a reversibility criterion of dynamic processes?
Dear Charles,
How do you prove that the second law is not more fundamental?and what is the exact initial condition?
I agree with the view that “the time symmetry of dynamical laws is just that, a mathematical symmetry in the equations. It does not mean that we can make things go backwards in time.” But more important is another sense of which in physics. Can a mirror image of T-symmetry of dynamical process be an actual process that nature allows? Yes means reversibility, and No means irreversibility.
T-symmetry of the first law of thermodynamics is only a mathematical symmetry in the equations, not means that a mirror image of T-symmetry of thermodynamic process can be an actual process in that the second law (T- symmetry broken), in thermodynamics, some T-mirror image process are prohibited by the second law, and had nothing to do with “the initial condition” and “overwhelming probability”.
I rewritten the statement about the second law, and you can also get it from thermodynamic equilibrium conditions:
“All of the gradients of the thermodynamic forces spontaneously tend to zero.”
The gradients of the thermodynamic forces include
1) “the temperature gradient”,
2) “the generalized force gradient”,
3) “the chemical potential gradient”
4) “the diffusive force gradient”.
Where 1) and 4) involve “probability”, “the initial condition” and “overwhelming probability”. 2) and 3) involve determinism.
“the time symmetry of dynamical laws is a mathematical symmetry in the equations,” this is not a problem. The problem is: Can all mirror image of T-symmetry of dynamical processes be actual processes that nature allows? There has no general proof.
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