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

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))

• It would be useful to clarify your question by stating what you consider to be Loschmidt's paradox. I assume you mean the fact that thermodynamics is time asymmetric when the underlying laws of physics are symmetric. – Philip Gibbs - inactive Jan 25 '12 at 8:16
• Note that if CPT symmetry holds (and no violations are known) the existence of CP violation implies that the fundamental laws of physics are not fully time symmetric. – dmckee --- ex-moderator kitten Jan 25 '12 at 18:02
• @dmckee but note that in that case Loschmidt's paradox is just as much (or as little) of a problem as before, because every trajectory still has an "antitrajectory" that's effectively the same but with time reversed. The only difference is that the antitrajectory has the charge and parity reversed, as well as velocities. – Nathaniel Aug 4 '12 at 20:47
• Plus, and as Feynman noted, these high-energy violations of CP are not enough to explain the time asymmetry of the huge mass of phenomena around us that's fully explained (at least fundamentally and in principle) by electromagnetic interactions. – Emilio Pisanty Aug 5 '12 at 1:53

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.

• "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." it seems like time and knowledge of an entity embedded in the system (which performs and learns information) is getting towards an answer – propaganda Jan 25 '12 at 9:00
• has the statistical mechanics relationship between microscopic variables (unknown to an entity within the system) and macroscopic variables (some of which form the "knowledge" of an entity in the system) been combined with Kripke semantics to explain the arrow of time? – propaganda Jan 25 '12 at 9:04
• Feel free to vote me down if you think the scientific community all agrees with your solution to this paradox. – Philip Gibbs - inactive Jan 25 '12 at 9:48
• actually I had already voted you up because: the second bullet looks a lot like a different interpretation of what I have in mind, and because it is a summary of what different directions the scientific community explores on the topic, which was part of the question – propaganda Jan 25 '12 at 10:51
• I'm not sure it doesn't make sense to declare axioms in physics, although it does import some philosophical baggage. It's conceivable (and I think in practice this is the implicit position of most physicists, or at least theorists) that the laws of physics derive from some set of "natural" axioms, for example. – AGML Aug 2 '16 at 17:41

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.

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.

• I think Motl may be the only person I know who thinks that the H-theorem resolves this issue directly. – Philip Gibbs - inactive Jan 25 '12 at 9:34
• Could it not be that this axiom is not independent from more fundamental axioms (from say logic), i.e. it does not harm to add it as an axiom as it wont conflict, but it may be a corrolary of widely used deeper axioms, of which we dont know the proof yet? – propaganda Jan 25 '12 at 9:38
• Otherwise your games with the "definition" of initial and final states and with the sign of $t$ are completely immaterial. "Initial" and "final" states are, according to logic, qualitatively different things, and the usual convention for the sign of $t$ is that $t_{\rm initial}<t_{\rm final}$. But I have never even used this convention. Even if I had, it wouldn't matter. One can easily rewrite all proofs to the opposite convention by replacing $t$ with $-t$; all those things are physically vacuous. The non-vacuous claim is that the future and past don't play symmetric roles in logic. – Luboš Motl Jan 25 '12 at 9:49
• Yes, @MarcoDisce, it's the transition from the "everything is known" deterministic classical mechanics to statistical physics (or quantum mechanics) which breaks the time-reversal symmetry. The point of statistical physics is that there's some uncertainty about details or, equivalently, that microstates are clumped to groups, ensembles. And the laws for ensembles are asymmetric. The probability of ensemble A becoming B in the future is the sum over probabilities of microstates in B, the final state, but the average over the initial state. Average and sum differ by 1/N, asymmetry... – Luboš Motl May 27 '16 at 8:33
• But the formulae needed to predict the future and to retrodict the past are completely different. Moreover, the retrodiction always depends on arbitrary "prior probabilities" (the odds are not fully calculable from the laws of physics) because it's a form of Bayesian inference. The reason of the asymmetry is that the laws of physics specify the numerical values of P(F|I), the conditional probability of the final state given the initial state, and it's not the same thing as P(I|F) once probs between 0 and 100% enter. – Luboš Motl May 27 '16 at 13:33

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.

• I think this excellent answer shows that the OP, and physicists in general have to be very careful of exactly what they mean by "paradox". Maybe a better word would be "mystery". As Nathaniel says, E. T. Jaynes looked at this problem very deeply and gave a highly plausible reformulation of the mystery. There are still questions to answer, but Jaynes's work narrows these down and clarifies them. To some people, "paradox" is very strong - almost synonymous with "logical contradiction", however the OED itself cites physics: – Selene Routley Jul 19 '13 at 6:36
• "the uncertainty principle leads to all sorts of paradoxes, like the particles being in two places at once" - i.e. "paradox" is simply a synonym for odd and intriguing - or, otherwise put, grounds for employment of physicists! It is unfortunate that some "paradoxes" spoken of in science are genuine logical contradictions (Russell Paradox), others are "removable" contradictions (owing to ambiguity, which when recognised removes the contradiction)(Berry Paradox) whilst others simply label something interesting (Twin Paradox) or, as here, quite sound but needing further explanation. – Selene Routley Jul 19 '13 at 6:44
• "This then leaves open the question of why our ability to interact with physical systems is time-asymmetric in this way". Actually this is what we think we can do just because our memory accumulation process is time-asymmetric. If we were in a "Loschmidt's reversal" or a "Poincarè's return" to low entropy the direction of our memory accumulation would be reversed. – Marco Disce May 27 '16 at 7:38

• "The Loschimidt paradox does not state that reversible laws of motion can not imply irreversible processes which sounds like a philosophical objection." Actually it would be a mathematical objection: a dynamical system that has time-symmetric evolution cannot produce a time-asymmetric system when we consider macroscopic variables that are time independent function of the microscopic variable (which have a time-symmetric evolution) – Marco Disce May 27 '16 at 7:54
• Maybe there's a connection with quantum indeterminacy. The reversal of a sequence of events is only required to return to the initial conditions if the process is deterministic. If the sequence wasn't deterministic, if there was quantum randomness in the evolution of the system, then the reversal isn't guaranteed to recover the initial state. I wonder what this says about entropy. – David Elm Nov 24 '16 at 15:06
• Not sure, but hasn't that part of the paradox resolved by agreeing that the molecular chaos assumption made by Boltzmann in his derivation was not correct and the velocities had correlations because of which you couldn't write them as one variable function product? – Normie Mar 12 '20 at 14:10

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.

• Could you please illustrate or point to the specific derivation/formulae which show that "its solution depends exponientally on the initials conditions."? – propaganda Aug 13 '12 at 4:54
• It is in one of those books. The demonstration is pretty long. Look first into: Radu Balescu: EQUILIBRIUM AND NONEQUILIBRIUM STATISTICAL MECHANICS : John Wiley & Sons, New York, 1975 Or into the vol 1 of those: Radu Balescu: TRANSPORT PROCESSES IN PLASMAS Vol. 1: CLASSICAL TRANSPORT Vol. 2: NEOCLASSICAL TRANSPORT – Shaktyai Aug 13 '12 at 6:22

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.

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.