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> I'm not sure I understand what you mean by "those microscopic processes that are compatible with macroscopic processes that systematically lead to the same fi

by effie 4y ago

> I'm not sure I understand what you mean by "those microscopic processes that are compatible with macroscopic processes that systematically lead to the same final state". (The same as what? The same in what sense?) In the formulation of 2nd law for adiabatically isolated systems, there is the initial equilibrium macrostate A, and the different equilibrium macrostate B that the system gets into eventually after the intervention, where either some constraint is released, or some macroscopic work is done. 2nd law states that if the same intervention in A results in B reliably, then S_B >= S_A. This statement can be derived from mechanics and maximum entropy principle only under the assumption that B is a reliable result of intervention in A (always or with close to 1 probability). This is more than mechanics or logic can provide, and it means we have a restriction on the allowed set of microscopic processes considered. It is not possible to derive 2nd law from mechanics or logic for all microscopic processes. Some microscopic processes bring the state B to state A, and those break 2nd law.

lisper 4y ago
> the initial equilibrium macrostate A, and the different equilibrium macrostate B that the system gets into eventually [emphasis added] Your use of "the" here, with the implication that there are two unique microstates A and B, is problematic. There is no such thing as "the" equilibrium microstate. A system at thermodynamic equilibrium is not static. (In fact, I think it is fair to say that the whole point of thermodynamics is to sweep vast numbers of the physical degrees of freedom of a system under the rug so that you can treat a system at equilibrium as if it were static even though it really isn't. It is frankly astonishing that this is even possible at all without losing any fidelity in terms of measurable outcomes.) > Some microscopic processes bring the state B to state A, and those break 2nd law. Yes, that's true. The second law is probabilistic. It is "possible" to break the second law in the same sense that it is "possible" for a baseball to quantum-mechanically tunnel through a catcher's mitt. You can do the math to figure the odds in both cases. In fact, actually going through this process is an informative exercise. It isn't difficult. There are some events whose probabilities are so low that the chance of them occurring anywhere in the universe before heat death is barely distinguishable from zero. The odds of actually observing such an event here on earth are vastly smaller still. Both macroscopic tunneling and 2nd law violations are events of this sort. Neither is categorically impossible, but (to put it mildly) it's pretty safe to bet against them nonetheless.
- effie 4y ago
  You have missed my point. The formulation of 2nd law I'm talking about is stating something about all pairs of two MACROstates A,B, not MICROstates, such that B reliably succeeds A after intervention. This pair of MACROstates is not assumed unique, there are hugely many, but formulating the argument uses a single pair. (I'm not a native speaker so don't get distracted by possibly incorrect use of the definite article). The problem with your handwavy argument about macrostates with higher multiplicity is not due to 2nd law being a probabilistic claim. The problem is that it is not clear from that argument alone why it applies to future states, but it does not apply to the past states. To address that, another assumption is needed, such as the asymmetric relation between my macrostates A,B.
  - lisper 4y ago
    Sorry, my mistake, I somehow misread "macro" as "micro". More than once. But it was obviously an "a" all along because it's an "a" in my quote. I need to start paying closer attention. So, having now re-read what you wrote upstream more carefully, can you elaborate on what you mean by "intervention"? > The problem is that it is not clear from that argument alone why it applies to future states, but it does not apply to the past states. I don't understand this. What do you mean that it "does not apply to past states"? > another assumption is needed, such as the asymmetric relation between my macrostates A,B Well, yeah, there is an asymmetric relationship: one precedes the other, and not the other way around. But that's not an "assumption", that is just how you defined A and B. I really don't see the problem you are trying to describe.
    - effie 4y ago
      By intervention I mean some external agent causing the system to leave the initial equilibrium macrostate A, which would not happen spontaneously. E.g. a wall is removed, or a piston is quickly pushed, to make the system evolve towards new equilibrium macrostate. > given a set of measurement values, the more states there are which produce those values, the more likely we are, all else being equal, to observe those values. And so a closed system will evolve by pure chance into a state where the entropy is maximized simply because you are more likely to observe values corresponding to greater numbers of underlying states. This argument is that future macrostates of an isolated system passing through non-equilibrium states (due to some intervention to the system that was in an equilibrium state before) should be almost always of higher entropy, because those have immensely higher multiplicity than macrostates with lower entropy. There is hugely more microstates that are compatible with higher entropy macrostate, than microstates that are compatible with lower entropy macrostate. However notice the word "future". We can formulate the same argument using "past": past macrostates of an isolated system passing through non-equilibrium states should be almost always of higher entropy, because those have immensely higher multiplicity than macrostates with lower entropy. There is hugely more microstates that are compatible with higher entropy macrostate, than microstates that are compatible with lower entropy macrostate. Now, why does the argument give correct answer when applied to future macrostates but not when applied to past macrostates? We need additional assumption about the non-equilibrium process that makes the argument invalid for the past.

by effie 4y ago

lisper 4y ago

> the initial equilibrium macrostate A, and the different equilibrium macrostate B that the system gets into eventually [emphasis added] Your use of "the" here, with the implication that there are two unique microstates A and B, is problematic. There is no such thing as "the" equilibrium microstate. A system at thermodynamic equilibrium is not static. (In fact, I think it is fair to say that the whole point of thermodynamics is to sweep vast numbers of the physical degrees of freedom of a system under the rug so that you can treat a system at equilibrium as if it were static even though it really isn't. It is frankly astonishing that this is even possible at all without losing any fidelity in terms of measurable outcomes.) > Some microscopic processes bring the state B to state A, and those break 2nd law. Yes, that's true. The second law is probabilistic. It is "possible" to break the second law in the same sense that it is "possible" for a baseball to quantum-mechanically tunnel through a catcher's mitt. You can do the math to figure the odds in both cases. In fact, actually going through this process is an informative exercise. It isn't difficult. There are some events whose probabilities are so low that the chance of them occurring anywhere in the universe before heat death is barely distinguishable from zero. The odds of actually observing such an event here on earth are vastly smaller still. Both macroscopic tunneling and 2nd law violations are events of this sort. Neither is categorically impossible, but (to put it mildly) it's pretty safe to bet against them nonetheless.

effie 4y ago
You have missed my point. The formulation of 2nd law I'm talking about is stating something about all pairs of two MACROstates A,B, not MICROstates, such that B reliably succeeds A after intervention. This pair of MACROstates is not assumed unique, there are hugely many, but formulating the argument uses a single pair. (I'm not a native speaker so don't get distracted by possibly incorrect use of the definite article). The problem with your handwavy argument about macrostates with higher multiplicity is not due to 2nd law being a probabilistic claim. The problem is that it is not clear from that argument alone why it applies to future states, but it does not apply to the past states. To address that, another assumption is needed, such as the asymmetric relation between my macrostates A,B.
- lisper 4y ago
  Sorry, my mistake, I somehow misread "macro" as "micro". More than once. But it was obviously an "a" all along because it's an "a" in my quote. I need to start paying closer attention. So, having now re-read what you wrote upstream more carefully, can you elaborate on what you mean by "intervention"? > The problem is that it is not clear from that argument alone why it applies to future states, but it does not apply to the past states. I don't understand this. What do you mean that it "does not apply to past states"? > another assumption is needed, such as the asymmetric relation between my macrostates A,B Well, yeah, there is an asymmetric relationship: one precedes the other, and not the other way around. But that's not an "assumption", that is just how you defined A and B. I really don't see the problem you are trying to describe.
  - effie 4y ago
    By intervention I mean some external agent causing the system to leave the initial equilibrium macrostate A, which would not happen spontaneously. E.g. a wall is removed, or a piston is quickly pushed, to make the system evolve towards new equilibrium macrostate. > given a set of measurement values, the more states there are which produce those values, the more likely we are, all else being equal, to observe those values. And so a closed system will evolve by pure chance into a state where the entropy is maximized simply because you are more likely to observe values corresponding to greater numbers of underlying states. This argument is that future macrostates of an isolated system passing through non-equilibrium states (due to some intervention to the system that was in an equilibrium state before) should be almost always of higher entropy, because those have immensely higher multiplicity than macrostates with lower entropy. There is hugely more microstates that are compatible with higher entropy macrostate, than microstates that are compatible with lower entropy macrostate. However notice the word "future". We can formulate the same argument using "past": past macrostates of an isolated system passing through non-equilibrium states should be almost always of higher entropy, because those have immensely higher multiplicity than macrostates with lower entropy. There is hugely more microstates that are compatible with higher entropy macrostate, than microstates that are compatible with lower entropy macrostate. Now, why does the argument give correct answer when applied to future macrostates but not when applied to past macrostates? We need additional assumption about the non-equilibrium process that makes the argument invalid for the past.