Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution?

Question

Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution?

I have never seen dissipation explained, although what I have seen a lot is descriptions of dissipation (i.e. more detailed pathways/mechanisms for specific systems). Typically one introduces axioms of dissipation for example:

entropy $S(t_1) \geq S(t_0) \Leftrightarrow t_1 \geq t_0$ (most often in words)

These axioms (based on overwhelming evidence/observations) are sadly often considered proofs. I have no problem with useful axioms (and I most certainly believe they are true), but I wonder if it can be proven in terms of other (deeper and already present) axioms. I.e. is the axiom really independent? or is it a corollary from deeper axioms from say logic (but not necessarily that deep).

(my opinion is that a proof would need as axioms some suitable definition of time (based on connection between microscopic and macroscopic degrees of freedom))

Answer 1

Loschmidt's paradox is that the laws of thermodynamics are time asymmetric because entropy always increases, but the underlying laws of physics are symmetric under time reversal. It should not therefore be possible to derive the second law of thermodynamics from first principles. Opinions in the scientific community differ as to whether this has been resolved (which implies that it has not been resolved) One commonly held opinion is that entropy increases only because it was low at the big bang, but that we don't know why it had to be low at the beginning. There are other possible explanations some of which also have significant support.

One point to make is that physics is not about axioms and proofs. These belong to mathematics which can be used to understand physical models and theories, but it makes no sense to declare axioms for physics. Any model must be tested against experiment and nothing is as absolute in science as an axiom. Thermodynamics in particular is a statistical science so its laws may only apply in closed systems of many degrees of freedom moving between states of equilibrium.

Some people still think that Boltzmann's H theorem explains why entropy always increases, but as Loschmidt's paradox implies, it must have a hidden time asymmetric assumption to work. You cannot get asymmetric solutions from symmetric equations unless there is a mechanism of spontaneous symmetry breaking (which the H theorem does not have) Boltzmann assumed that the initial state has low entropy and that there is nothing to constrain the future states to have low entropy. This leaves open the question as to why the initial state of the universe had low entropy. Since we do not yet have a complete theory of the initial state of the universe we cannot expect to be able to answer this question yet.

There are other ways that the paradox might be resolved with varying degrees of support from physicists. Here are three of them:

• CPT is most likely an exact symmetry of nature but CP and T are not. It could be that this small asymmetry drives the second law of dynamics by leaving the universe dominated by matter rather than anti-matter.

• It could be that the time asymmetry of the universe is driven by the laws of quantum mechanics through the measurement process which is time-asymmetric.

• In the theory of eternal inflation spacetime is always expanding. This is itself time-asymmetric and could be considered as a mechanism of spontaneous symmetry breaking that drives the arrow of time.

Answer 2

I think most people would say the paradox is resolved - but, as the answers to this question make clear, they wouldn't necessarily agree about who resolved it or what precisely the resolution is. For my money the paradox was elegantly resolved by Edwin Jaynes in this 1965 paper. In Jaynes' argument, the symmetry is broken by the fact that we, as experimenters, have the ability to directly intervene in the initial conditions of an (isolated) system, but we can only affect the final conditions indirectly, by changing the initial conditions.

Of course, this then leaves open the question of why our ability to interact with physical systems is time-asymmetric in this way. This is not a paradox but rather a physical fact in need of explanation. So while the mystery is not entirely solved by Jaynes' argument, at least the aparrent paradox can be laid to rest.

Answer 3

First of all, it's strange how the OP jumps from the Loschmidt "paradox" to dissipation. It makes it very unclear what he or she is actually asking because dissipation has no direct relationship to the Loschmidt "paradox" except that both of them are issues concerned with irreversibility in statistical physics or thermodynamics. The existence of dissipation is indisputable and demonstrable and all axioms or non-axioms in physics have to agree with this existence.

Irreversibility "paradox"

The Loschmidt "paradox" was an objection that Johann Loschmidt raised against (his younger colleague) Ludwig Boltzmann's claims about the statistical origin of entropy. In particular, Loschmidt claimed that Boltzmann shouldn't be able to prove the H-theorem – the increasing nature of entropy, a mathematical incarnation of the second law of thermodynamics (which implies a future-past asymmetry, the so-called thermodynamic arrow of time) – from microscopic laws that are invariant under the time reversal.

However, as Boltzmann understood, the objection is really invalid because all probabilistic reasoning in physics inevitably depends on the so-called logical arrow of time – which really says that the future is (fully or statistically but predictably) determined by the past but not in the other way around. For example, it follows from pure logic applied to events in time that if there are $N_0$ initial microstates and $N_1$ final microstates, the probability to get from the initial ensemble to the final ensemble must be averaged over the initial microstates but summed over the final microstates.

This really follows from pure logic; no other physical assumption is needed. We sum the probabilities over final states because we don't care which of them will occur and $P(A{\rm\,\,or\,\,} B)=P(A)+P(B)$ for mutually exclusive outcomes. We average the probabilities over the initial states because we don't know which of them was the right one and their prior probabilities have to satisfy $P(A)+P(B)+\dots = 1$. The asymmetry between the initial (past) state and the final (future) state doesn't depend on any details of the dynamics; it's pure logic. The logical arrow of time. It boils down to the asymmetry that the assumptions about the past and the claims about the future play in the Bayes formula. Implications in logic, $A\Rightarrow B$, aren't symmetric in $A,B$.

Note that the transition probability is therefore $$ {\rm Prob} = \sum_{i=1}^{N_0} \sum_{f=1}^{N_1} \frac{1}{N_0} {\rm Prob} (i\to f) $$ The factors $N_0$ and $N_1$ enter asymmetrically. The very fact that only $1/N_0$ is added is the reason why the evolution prefers a higher number of final states relatively to the initial states. One may compute the probability of the time-reverted process (or CPT-reverted process, to be more precise in QFT), and the factor will be $1/N_1$ instead. The ratio of these probabilities is therefore $N_1/N_0$ which is $\exp[(S_1-S_0)/k]$: and this ratio of probabilities which is extremely large for any macroscopic system guarantees that only the evolution in the direction where the entropy is increasing may occur with a detectably nonzero probability; the reverted process is impossible. Even though some people don't understand it, the rules for retrodiction are completely different from the rules for prediction: retrodiction is a form of (Bayesian) inference that, unlike predictions, always depends on (to some extent) arbitrary and subjective priors. Some people are making retrodictions according to the rules that only hold for predictions – and then they are surprised that they end up with absurd conclusions.

Ludwig Boltzmann organized the proof differently but he understood very well that his proof was actually a proof that the thermodynamic arrow of time is inevitably correlated with the logical arrow of time. People discovered quantum mechanics and lots of new reformulations of these arguments and proofs were written down but the essence hasn't changed. All physicists who understand and take statistical physics seriously understand that the Loschmidt "paradox" was already resolved by Boltzmann and there is no paradox. But much like 100 years ago, there exist people who don't understand the logic behind similar proofs in statistical physics and who keep on repeating misconceptions that there exists a Loschmidt "paradox". This is a purely social phenomenon that will probably not go away; 100 years ago, physics has simply become so advanced and abstract that most people, even those who manage to get "some" physics education, are already unable to get to the cutting edge (and even "not so cutting edge"). The situation is even more striking in the case of quantum mechanics.

At any rate, the relevant answer is that the competent part of the scientific community (especially most of the people who are statistical physics experts) agrees that the Loschmidt "paradox" was already addressed and resolved more than 100 years ago while a broader "community" is split about this issue.

Answer 4

The Loschimidt paradox does not state that reversible laws of motion can not imply irreversible processes which sounds like a philosophical objection. It rather observes that Boltzmann H-Theorem leads to the following physical contradiction: Take a system that starts at H_1 and evolves to H_2 and finally to H_3. The theorem states that H_3 < H_2 < H_1. Now take the microstate which correspond to H_2 and reverse the direction of all the velocities. We should all agree on the fact that at that point we would observe the system going back to H_1. Unfortunately the H-theorem states that the system will go to H_3 regardless of our intervention on the velocities. This is does not make sense at all, and this is why Loschmidt paradox is a real paradox, and not a solved paradox. A solved paradox is not a paradox. The reaction of Boltzmann indeed was not to try convince anyone that this paradox can be solved. His reaction was to leave the H-theorem in favor of a new prospective based on the combinatorial argument. Consider the classic Gibbs book for instance; you don't find anything similar to the H-theorem in his theory. What you find instead is the observation that in order to describe irreversible processes, you need to ignore the nature of mechanics expressed by the Liouville Theorem, and you need to introduce some different approach based on the coarse grain.. which is the same idea that Boltzmann had after Loschmidt objection.

Answer 5

Time asymetry appears in the solution of the Boltzmann's equation because its solution depends exponientally on the initials conditions. After a few caracteristic relaxation times, the initial conditions becomes exponentially small. So, although the microscopic particles obey Hamiltonian dynamics (with trajectories depending on initial conditions), as a whole this Hamiltonian caracteristic disappears and a new dynamics appears which for neutral gas is well modelized by the Boltzmann's equation. It is fundamental to understand that in statistical physics one can not think in terms of a single test particle. A single particle is a set with zero mesure which is irrelevant. There is a more problematic theorem: the Poincare's theorem which roughly states that any mechanical system goes back to its initial state. However, the time it takes to do it is for large system far greater than the age of the universe.

Answer 6

Although summarized as an objection of macroscopic irreversibility when microscopic laws are reversible, Loschmidt's objection originally points that there has to be something breaking the time reversal symmetry in Boltzmann's derivation of the $H$-theorem.

I think that Boltzmann's answer was to say that high $H$ states (in absence of external driving) are more the exception than the rule. This is betrayed by the fact that inverting time in the $H$-theorem still leads to a decrease in $H$.

I think it is important to stress that Boltzmann's equation (from which derives the $H$-theorem) only looks at a very coarse grained quantity, namely the one-particle density and most rationals for the asymmetry are put at this coarse grained level.

Yet, mathematicians are still working on the problem (see here and there ).

But as a physicist, and for a picture beyond physics of gases, I think that this article on relevant entropies gives a lot of insights about these things in general.

Answer 7

As others have noted, there is no agreement, but I don't find a problem on that because there are many other fields of physics where there is no agreement among scientists. :-D

Most of the so-named resolutions of the paradox are invalid. Concretely, the three 'explanations' mentioned on the Wikipedia article you mention are incorrect. The transfer operator method is based in an early approach developed by the Brussels-Austin School and latter abandoned by them, because a rigorous spectral decomposition provides two semigroups and two sets of eigenvalues (one of them compatible with the second law and other incompatible). Then the compatible set is chosen and the incompatible set rejected, but as shown latter by the School this is equivalent to breaking the time-symmetry of the microscopic laws and replacing an unitary evolution by a non-unitary law. This is the reason why in latter years the Brussels-Austin School developed a generalization of mechanics with a microscopic dissipation rule $\Im(\Theta) \le 0$ that could provide a rigorous foundation to the eigenvalue set that agrees with observations.

The fluctuation-theorem 'resolution' repeats the same uncritical mixing of mechanics and probabilistic aspects found in Boltzmann works. To not mention that the 'resolution' is based in a confusion between time-symmetry and microscopic reversibility, and produces well-known paradoxes and disagreement with observations due to introduction of anthropomorphic aspects like coarse-grained descriptions and the non-mechanical probabilities associated to them.

The theorem is also often used to compute the probability of a violation of the second law by people that believes that the second law is only "probabilistic" and valid "most of time" but not always. This is based in a misunderstanding of basic thermodynamics concepts. The second law is a statement about the average entropy $S$, not about the fluctuating entropy $\tilde{S}= S +\delta S$. The production of the average entropy has to be non-negative by the second law $\mathrm{d}_i S \ge 0$, the production of the second can be positive, negative, or zero depending of the random disturbances. A spontaneous reduction on entropy in a system due to molecular fluctuations doesn't violate the second law.

The cosmological 'resolution' pretends that the origin of the reversibility is on the initial conditions --initial cosmological state of low entropy--. This is another misunderstanding of thermodynamics. Consider the classical version of the second law for isolated system: $\Delta S \ge 0$. This can be written in the alternative form

$$ S(t) \geq S(t_0).$$

The second law doesn't make any further statement about the initial value of the entropy; $S(t_0)$ can be low, can be zero, or can be very very large. The second law doesn't say that the initial state has to be one of very low entropy. In fact, the second law also applies to systems are initially in a state of very high entropy --near the maximum possible--; it is the reason why when we disturb infinitesimally a system at equilibrium, the system will return to equilibrium when the perturbation is switched-off. Note also that when the system is initially in the state of maximum possible entropy the second la states $S(t) = S(t_0)$.

The cosmological 'resolution' also ignores the crux of the paradox. If the universe is in an initial state with entropy $S(t_0)$, the laws of mechanics (both classical and quantum) state that the entropy at any other time $S(t)$ will be the same, independently of which is the value of the entropy at the initial instant. Imagine that initial entropy is the lowest possible $S(t_0)=0$, then mechanics affirms that $S(t)=0$ for any other time. in disagreement with both the second law and observation.

The only possible resolution of the paradox consists on formulating irreversible microscopic equations. There a large literature in the topic and different proposals. The XXI Solvay Conference on Physics discussed some of those proposals.

Answer 8

1. The Second Law has nothing to do with time, and it isn't formulated in terms of S. Instead, it is used to define entropy. See Caratheodory and Born.

2. Loschmidt's objection concerns the trivial fact: irreversible macroscopic behaviour cannot be derived using reversible equations of motion. Boltzmann obviously hailed. His H-theorem is just a mathematical exercise. Later, Ehrenfest attempted to do the job using corse-graining of the phase space, but also failed.

3. The reversibility problem isn't properly defined. What is irreversible is thermodynamic state. A mechanical state is perfectly reversible by time inversion in the equations of motion. But a thermodynamic state is not. Any adiabatic transformation is adiabatically irreversible.