A Derivation of Statistical Mechanics

[gentzen]

The fine grained probability density $\rho(x)$ satisfies the continuity equation if and only if $\rho(x) +|\Psi(x)|^2$.

That's a nonsense. As long as particles are not created or destructed (which are not in BM), the continuity equation is satisfied for any $\rho(x)$.

But the current in such a continuity equation is not given as a function of the wavefunction and its derivatives.

More precisely, for any initial $\rho(x,0)$ there is $\rho(x,t)$ such that $\rho(x,t)$ satisfies a continuity equation.

That was a later addition in an edit, which intentionally wrote "a continuity equation" instead of "the continuity equation". Indeed, I agree that you are right in your defense of coarse graining for BM. But since Nullstein is nicely focused in his points, I think I am right too in making the concrete difference between "a continuity equation" and "the continuity equation" more explicit.

 

[Demystifier]

But the current in such a continuity equation is not given as a function of the wavefunction and its derivatives.

That was a later addition in an edit, which intentionally wrote "a continuity equation" instead of "the continuity equation". Indeed, I agree that you are right in your defense of coarse graining for BM. But since Nullstein is nicely focused in his points, I think I am right too in making the concrete difference between "a continuity equation" and "the continuity equation" more explicit.

Well, if Nullstein meant this by the continuity equation, then he is even more wrong. He said that "Otherwise, probabilities won't add up exactly to 1 and can't be interpreted as probabilities anymore.". But probabilities add up to 1 for any $\rho(x,t)$ that satisfies (i) the initial condition $\int d^3x \, \rho(x,0)=1$ and (ii) continuity equation for all times with this $\rho(x,t)$.

 

[Nullstein]

That's a nonsense. As long as particles are not created or destructed (which are not in BM), the continuity equation is satisfied for any $\rho(x)$. More precisely, for any initial $\rho(x,0)$ there is $\rho(x,t)$ such that $\rho(x,t)$ satisfies a continuity equation.

You shouldn't just call something nonsense just because you don't understand it. In QM, the continuity equation is given by $\partial_t|\Psi|^2 +\nabla\vec j = 0$, where $\vec j$ is the probability current derived from the Schrödiger equation. In BM, one defines the velocity of the hidden variables to be $\vec v = \frac{\vec j}{|\Psi|^2}$ and you can insert it into the conuity equation to arrive at $\partial_t|\Psi|^2 +\nabla(|\Psi|^2\vec v)=0$. That's fine, because $|\Psi|^2\vec v = \vec j$ and the equation still holds, because it's just a reformulation. But if you now replace $|\Psi|^2$ by some different function $\rho$ in that equation, the equation will no longer be satisfied.

It can never be a mistake to just work with the fine grained distribution. If there is a coarse grained distribution that makes the same predictions, this is fine, but if it makes different predictions than the fine grained version, then using the coarse grained version is just a mistake. In this case, the above discussion proves that the fine grained distribution will not satisfy the continuity equation if it is not equal to $|\Psi|^2$, which leads to probabilitites that don't add up to $1$, so the fine grained distribution will cease to be a probability distribution.

 

[Nullstein]

Well, if Nullstein meant this by the continuity equation, then he is even more wrong. He said that "Otherwise, probabilities won't add up exactly to 1 and can't be interpreted as probabilities anymore.". But probabilities add up to 1 for any $\rho(x,t)$ that satisfies (i) the initial condition $\int d^3x \, \rho(x,0)=1$ and (ii) continuity equation for all times with this $\rho(x,t)$.

You can't just use any random continuity equation. There is exactly one fine grained distribution, which satisfies one specific continuity equation, which follows from the Schrödinger equation. You are free to use a coarse grained version of the distribution if its predictions agree with the fine grained version (up to some precision). But in this case, it is even provable that the fine grained distribution won't even remain a probability distribution after some time. Using the fine grained distribution can never be wrong. Using the coarse grained version can only be convenience. If the fine grained version tells you that probabilities don't add up to $1$, then you have a serious problem.

 

[gentzen]

You can't just use any random continuity equation. ... But in this case, it is even provable that the fine grained distribution won't even remain a probability distribution after some time.

This seems wrong to me. For the fine grained distribution to remain a probability distribution, it is enough that is remains non-negative, and follows any random continuity equation.

 

[Nullstein]

This seems wrong to me. For the fine grained distribution to remain a probability distribution, it is enough that is remains non-negative, and follows any random continuity equation.

You're right, it's not necessarily provable that it ceases to be a probability distribution, but it may. There is only one continuity equation that is derived from the underlying physics and that's the one derived from the Schrödinger equation. It certainly won't satisfy this one.

 

[Nullstein]

Another way to see that the use of Gibbs' H-theorem is invalid is the following: Gibbs' H-theorem really evolves every initial distribution into equilibrium. If that was reasonable, then on a macroscopic scale, everything would seem to be in equilibrium. There would be no rivers, no storms, no solar systems and so on. There would not be non-equilibrium stationary states. Gibbs' H-theorem even evolves fully integrable systems into equilibrium, which can't possibly be argued to spread over phase space.

 

[Demystifier]

You're right, it's not necessarily provable that it ceases to be a probability distribution, but it may.

No, it may not. It's provable that any continuous deterministic law for velocities with any initial distribution of positions gives rise to a continuity equation.

There is only one continuity equation that is derived from the underlying physics and that's the one derived from the Schrödinger equation.

No, Schrodinger equation alone does not imply that probability is given by $|\psi|^2$. You must assume something more to derive that probability is equal to $|\psi|^2$. That's, indeed, a big problem for the many-world interpretation.

 

[Nullstein]

No, Schrodinger equation alone does not imply that probability is given by $|\psi|^2$.

I didn't claim that though. I just claimed that Schrödinger's equation implies the continuity equation $\partial_t|\Psi|^2+\nabla\vec j = 0$ with the standard Schrödinger current $\vec j$. It doesn't imply any other continuity equation.

 

[Demystifier]

Gibbs' H-theorem really evolves every initial distribution into equilibrium.

No it doesn't. It evolves most initial conditions close to equilibrium.

If that was reasonable, then on a macroscopic scale, everything would seem to be in equilibrium.

No it wouldn't, because time needed to come close to equilibrium can be very long. For some estimates see https://en.wikipedia.org/wiki/Heat_death_of_the_universe#Timeframe_for_heat_death

 

[Demystifier]

I didn't claim that though. I just claimed that Schrödinger's equation implies the continuity equation $\partial_t|\Psi|^2+\nabla\vec j = 0$ with the standard Schrödinger current $\vec j$. It doesn't imply any other continuity equation.

That's both wrong and irrelevant.

It is wrong because any other $\vec j'$ of the form
$$\vec j'=\vec j+\vec u,$$
where $\vec j$ is the standard Schrodinger current and $\vec u$ is an arbitrary vector field satisfying $\nabla \vec u=0$, also satisfies the continuity equation $\partial_t|\Psi|^2+\nabla\vec j' = 0$.

It is irrelevant because the continuity equation above only implies that $\rho=|\Psi|^2$ is consistent, it does not imply that $\rho=|\Psi|^2$ is necessary.

 

[Nullstein]

No it doesn't. It evolves most initial conditions close to equilibrium.

Yes, it does. He proves that after each coarse graining step, the entropy increases, without further qualifications. Since, there is a maximum entropy, every state is evolved into a maximum entropy state and therefore into equilibrium. There are non-equlibrium stationary states, i.e. states that are invariant under the Liouville dynamics. Even those states are evolved into equilibrium by Gibbs' H-theorem. The blurring in Gibbs' H-theorem is an irreversible, entropy increasing process.

No it wouldn't, because time needed to come close to equilibrium can be very long. For some estimates see https://en.wikipedia.org/wiki/Heat_death_of_the_universe#Timeframe_for_heat_death

Gibbs' H-theorem evolves everything quickly into equilibrium, it's disconnected from the physical dynamics. Sure, the universe will thermalize, but not because of Gibbs' H-theorem. But even if it would take a long time, then you argument could just equally well be applied to BM to argue that the probability distribution is not given by $|\Psi|^2$ for a very long time.

 

[Demystifier]

He proves that after each coarse graining step, the entropy increases, without further qualifications.

There are further qualifications. In Phys. Lett. A 156, 5 (1991), Valentini says (at the beginning of paragraph after Eq. (9)): "The proof rests on the assumption of no "micro-structure" for the initial state, as for the classical case."

 

[Demystifier]

Gibbs' H-theorem evolves everything quickly into equilibrium, it's disconnected from the physical dynamics. Sure, the universe will thermalize, but not because of Gibbs' H-theorem. But even if it would take a long time, then you argument could just equally well be applied to BM to argue that the probability distribution is not given by $|\Psi|^2$ for a very long time.

Various numerical simulations in classical and Bohmian mechanics show that (i) Gibbs equilibriation does depend on physical dynamics and (ii) equilibriation in BM is much faster due to nonlocal interactions.

 

[Nullstein]

There are further qualifications. In Phys. Lett. A 156, 5 (1991), Valentini says (at the beginning of paragraph after Eq. (9)): "The proof rests on the assumption of no "micro-structure" for the initial state, as for the classical case."

This is not a contradiction, you should have quoted the whole paragraph. He writes: "The proof rests on the assumption of no "microstructure" for the initial state, as for the classical case. Specifically, we assume the equality of coarse-grained and fine-grained quantities at the initial time t=0, i.e. we assume $\bar P(0)=P(0), \left<|\Psi(0)|^2\right>=|\Psi(0)|^2$". This is indeed one assumption in Gibbs' proof. But it's not an assumption of the kind you were indicating. Gibbs just assumes that at $t=0$, the coarse grained distribution is given by the fine grained distribution. But there is no further qualification on the fine grained state. Every fine grained state will be evolved into equilibrium by Gibbs' H-theorem.

Various numerical simulations in classical and Bohmian mechanics show that (i) Gibbs equilibriation does depend on physical dynamics

Yes, Gibbs' H-theorem is a consecutive application of Liouville dynamics, then blurring, then Liouville dynamics, then blurring and so on. The Liouville dynamics is the physical input. But the blurring is an artificial, entropy increasing process. Gibbs' H-theorem evolves every state into equilibrium, independent of the physical equations of motion. The physical dynamics are there to put the correct Hamiltonian in $e^{-\beta H}$.

(ii) equilibriation in BM is much faster due to nonlocal interactions.

Gibbs' H-theorem is always fast. The point is that Gibbs' H-theorem is unphysical, because the blurring is unphysical. The speed convergence is much faster than the actual dynamics (if there is equilibration in the actual dynamics in the first place). The universe may take $10^100$ years to thermalize, yet Gibbs' H-theorem will evolve it into equilibrium very quickly. Some systems (such as fully integrable systems) never equilibrate, yet Gibbs' H-theorem evolves them into equilibrium nevertheless. This just shows that Gibbs' H-theorem is inappropriate to decide whether something evolves into equilibrium or not.

 

[Demystifier]

Gibbs' H-theorem is a consecutive application of Liouville dynamics, then blurring, then Liouville dynamics, then blurring and so on.

I have already explained that blurring does not work that way. I used analogy with a strong computer and a low resolution screen.

 

[Nullstein]

I have already explained that blurring does not work that way. I used analogy with a strong computer and a low resolution screen.

Blurring in Gibbs' H-theorem works this way.

 

[Demystifier]

Blurring in Gibbs' H-theorem works this way.

Can you give a reference?

 

[Nullstein]

Can you give a reference?

See any proof of Gibbs' H-theorem, e.g. Tolman "Principles of Statistical Mechanics".

 

[Demystifier]

See any proof of Gibbs' H-theorem, e.g. Tolman "Principles of Statistical Mechanics".

I'm looking at Sec. 51 devoted to the Gibbs generalization of the Boltzmann's H-theorem. In this section I cannot find any confirmation of your claims. Can you be more specific by giving the exact page and/or exact quote?

 

[Nullstein]

I'm looking at Sec. 51 devoted to the Gibbs generalization of the Boltzmann's H-theorem. In this section I cannot find any confirmation of your claims. Can you be more specific by giving the exact page and/or exact quote?

He starts with a fine grained distribution $\rho_1$ and a distribution $\P_1=\rho_1$. He then evolves both using Liouville dynamics. $\rho_1$ evolves into $\rho_2$. $P_2$ arises by evolving $P_1$ using Liouville's dynamics and then restricting it to a coarse grained phase space. He shows that $\rho_2\neq P_2$ in eq. (51.16). He shows that the entropy of the coarse grained distribution increased in eq. (51.20). The entropy of the fine grained distribution remains constant (obviously, because Liouville dynamics is reversible). He then says that further time evolution of this kind will further increase the entropy of the coarse grained state at the end of p. 172, i.e. the the state needs to be evolved and coarse grained again in order to further increase entropy until $S_{max}$ is reached.

 

[Demystifier]

He starts with a fine grained distribution $\rho_1$ and a distribution $\P_1=\rho_1$. He then evolves both using Liouville dynamics. $\rho_1$ evolves into $\rho_2$. $P_2$ arises by evolving $P_1$ using Liouville's dynamics and then restricting it to a coarse grained phase space. He shows that $\rho_2\neq P_2$ in eq. (51.16). He shows that the entropy of the coarse grained distribution increased in eq. (51.20). The entropy of the fine grained distribution remains constant (obviously, because Liouville dynamics is reversible).

So far so good.

He then says that further time evolution of this kind will further increase the entropy of the coarse grained state at the end of p. 172, i.e. the state needs to be evolved and coarse grained again in order to further increase entropy until $S_{max}$ is reached.

The crucial question here is what one means by "the state". In my understanding it is the fine grained state (that is evolved and coarse grained again), while in your understanding it is the coarse grained state (that is evolved and coarse grained again). The text of Tolman does not seem explicit about that. Your understanding requires the coarse graining to be physical, while my understanding only requires a lack of information about the fine grained state. Tolman says that it "corresponds to a decrease with time in the definite character of our information" which indicates that his understanding coincides with mine, not with yours.

 

[Nullstein]

The crucial question here is what one means by "the state". In my understanding it is the fine grained state (that is evolved and coarse grained again), while in your understanding it is the coarse grained state (that is evolved and coarse grained again). The text of Tolman does not seem explicit about that. Your understanding requires the coarse graining to be physical, while my understanding only requires a lack of information about the fine grained state. Tolman says that it "corresponds to a decrease with time in the definite character of our information" which indicates that his understanding coincides with mine, not with yours.

Your understanding can be proven to be wrong quite easily: Suppose $\rho_1$ is a delta state, then its coarse grained version is just a uniform distribution over the phase cell where the system is located. The time evolution of this state will again be a delta state and its coarse gaining will again be a uniform distribution in a (different) phase cell. The entropy of a uniform distribution in a phase cell is given by some constant and depends only on the size of the phase cell, not on its location. Therefore, the entropy will not increase according to your understanding. And since a uniform distribution on a phase cell is not a thermal state, your version of this process won't evolve the state into an equilibrium and $S_{max}$ will never be reached. Your version of this process will only make the particle from one phase cell into another in the coarse grained picture. It will never spread across multiple phase cells.

 

[PeterDonis]

The entropy of a uniform distribution in a phase cell is given by some constant and depends only on the size of the phase cell, not on its location.

But the size of the phase cell the fine-grained "delta" state is in does depend on its location--as the fine-grained "delta state" evolves in time, it moves into different phase cells with different sizes--and those sizes are overwhelmingly likely to be larger than the size of the original phase cell. That's why the entropy increases (or more precisely is overwhelmingly likely to increase).

 

[Nullstein]

But the size of the phase cell the fine-grained "delta" state is in does depend on its location--as the fine-grained "delta state" evolves in time, it moves into different phase cells with different sizes--and those sizes are overwhelmingly likely to be larger than the size of the original phase cell. That's why the entropy increases (or more precisely is overwhelmingly likely to increase).

Not true. The phase cells are all of the size $\delta q\delta p\ldots$ with some fixed lengths $\delta q$, $\delta p$. It's just a lattice version of phase space with a fixed lattice spacing.

 

[PeterDonis]

It will never spread across multiple phase cells.

Who said it would?

 

[Nullstein]

Who said it would?

Demystifier. At least if he claims that the state approaches a thermal state. The coarsed grained version of $e^{-\beta H}$ certainly has contributions in all phase cells where $H\neq\infty$.

 

[PeterDonis]

Not true. The phase cells are all of the size $\delta q\delta p\ldots$ with some fixed lengths $\delta q$, $\delta p$. It's just a lattice version of phase space with a fixed lattice spacing.

I have no idea where you are getting this from. Coarse graining in thermodynamics means each phase space "cell" consists of all phase space points that correspond to the same values (or more precisely functions) for all macroscopic thermodynamic variables. Those cells will not all be the same size.

 

[Nullstein]

I have no idea where you are getting this from.

Read the book that I cited.

Coarse graining in thermodynamics means each phase space "cell" consists of all phase space points that correspond to the same values (or more precisely functions) for all macroscopic thermodynamic variables. Those cells will not all be the same size.

Coarse graining in Gibbs' H-theorem means projecting the fine grained state on a lattice version of phase space with fixed sized phase cells. Again, please read the book (specifically eq. (51.3)).

 

[Demystifier]

Suppose $\rho_1$ is a delta state, then its coarse grained version is just a uniform distribution over the phase cell where the system is located. The time evolution of this state will again be a delta state and its coarse gaining will again be a uniform distribution in a (different) phase cell.

It's true for the delta state, but not true for most other fine grained states.

Anyway, I have checked also the book by Jancel, "Foundations of Classical and Quantum Statistical Mechanics", Sec. "Discussion of the generalised H-theorem" starting at page 169. He starts with "It can be seen from the foregoing developments that this theorem is obtained without any special assumptions, except the fundamental assumption of statistical mechanics, which is necessary for the statistical description of macroscopic phenomena and which cannot contradict the reversibility of the laws of mechanics." Likewise, the section ends with: "... the evolution takes place according to the deterministic laws of mechanics, the irreversibility arising from the gross nature of our observations. Thus, it is not possible to have any kind of contradiction between the statistical conclusions of the theory and the reversible behaviour of mechanical systems." I think it contradicts your view.