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> You have not answered my question. You aren't getting it. Rising Entropy is just an example for illustrative purposes, entropy is simply a probabilistic pheno

by kgwgk 4y ago

> You have not answered my question. You aren't getting it. Rising Entropy is just an example for illustrative purposes, entropy is simply a probabilistic phenomenon. I thought that your question was how probability "appears" from classical mechanics and the answer is that you're putting it in - either by obligation (because you cannot describe the system perfectly) or by choice (because you decide not to do so even if you could). Entropy arises when the classical state of the system is not identified precisely as a point in phase space and you have a volume instead corresponding to the possible physical states. More uncertainty about the state, more volume, higher entropy. Less uncertainty about the state, less volume, lower entropy. No uncertainty about the state, no volume, zero entropy. >> Only if we keep adding uncertainty / forgetting information as we go forward will entropy increase. > Bro. Forgetting stuff doesn't change entropy. We already agreed that entropy is independent of knowledge. We already agreed that the entropy depends on the description that we make of the system. By forgetting stuff or otherwise adding uncertainty you get a worse description than what you could have if you didn't - that means higher entropy. If the uncertainty doesn't increase somehow the entropy remains constant. Entropy is a function of the volume in phase space and that volume is constant in classical -and quantum, for the matter- mechanics. If the equations describing the evolution are perfectly known the uncertainty doesn't have to increase. It doesn't matter how much time passes, the particles cannot be _anywhere_: their possible positions in the future are determined by their possible positions in the past. But this constant volume of possible positions may get more and more spread around even though they used to be in a compact region. See Figure 1.7 here: https://books.google.ch/books?id=DjUgfVFtY6YC&lpg=PA27&ots=aT3UBjrwTc&pg=PA27#v=onepage&q&f=false https://books.google.ch/books?id=DjUgfVFtY6YC&lpg=PA27&ots=a... By not making use of that additional information - possibly because we just can't afford that level of detail - we let uncertainty grow. That's what I meant by adding uncertainty / forgetting information. The volume in phase space representing the possible states is larger. The entropy of the degraded description of the system is higher.

kgwgk 4y ago
You said that you where thinking about Gibbs entropy. He came up with the ideas above. Including coarse-graining - the “add uncertainty losing the details to allow for entropy to increase” part. https://aardvark.ucsd.edu/grad_conference/hagar.pdf https://aardvark.ucsd.edu/grad_conference/hagar.pdf In Gibbs’ approach one represents the microstate of a physical system with N particles each with f degrees of freedom by a point X ∈ Γ where Γ, the phase space of the system, is a 2Nf-dimensional space spanned by the Nf momenta and Nf configuration axes. As the system evolves this representative point will trace out a trajectory in Γ which obeys Hamilton’s equations of motion. Next, one considers a fictitious ensemble of individual systems (represented by a ‘cloud’, or a ‘fluid’, of points on phase space) each in a microstate compatible with a given macrostate (say, such and such energy in such and such pressure contained in such and such volume). The macroscopic parameters thus pick out a distribution of points in Γ. We then ascribe a normalized density function to the ensemble, ρ(p,q,t), and, except for entropy and temperature, the mean value of phase function with respect to ρ describes the system’s thermodynamic properties.[…] All this is so very fine, but if Gibbs’ systems obey Hamilton’s equations of motion then the ‘cloud’ representing them in phase space swarms like an incompressible fluid. Consequently his ‘fine-grained’ entropy as defined in (7.9) is invariant under the Hamiltonian flow: dSF/dt G(ρ) = 0. (7.10) If there is a problem in SFG it is not just the fact that it does not move. Recall that TD entropy is defined only in equilibrium, so in order to construct a mechanical counterpart one only needs to find a function whose value at a later equilibrium state is higher than at an earlier equilibrium state.21 But since the macroscopic parameters change between the two equilibrium states, the Gibbs’ approach has no problem in doing this just by defining a new ensemble with a new probability distribution for the new equilibrium state and this will match the thermodynamic entropy as before. […] it is not fair to use the macro-parameters, which are supposed to be derived from the micro-parameters, in order to construct the latter. In other words, the ensemble at later equilibrium state should be the Hamiltonian-time-evolved ensemble of the earlier equilibrium state, otherwise the system is not governed by Hamilton’s equations as one originally presupposes. Thus, if one wants to use Gibbs’ fine-grained entropy as a mechanical counterpart to TD entropy, then one must abandon standard, Hamiltonian, dynamics since it does not connect the two fine-grained equilibrium states. That this is the true problem with Gibbs’ fine-grained entropy escaped many commentators, and as a result the foundations of SM were soon piled with a lot of dead wood. Stemming from the famous Ehrenfests’ paper (1912, 43–79) where Boltzmann’s students complained on Gibbs’ treatment of irreversibility by categorizing it bluntly as “incorrect”, the last century was consumed with attempts to find a monotonically increasing function as a counterpart for TD entropy. One way to achieve this goal is to follow Gibbs himself, who introduces the mathematical trick of ‘coarse graining’ and devises new notions of entropy and equilibrium. In this approach one divides Γ into many small finite cells of volume ω and then takes the average of ρ over these cells.

by kgwgk 4y ago

kgwgk 4y ago

You said that you where thinking about Gibbs entropy. He came up with the ideas above. Including coarse-graining - the “add uncertainty losing the details to allow for entropy to increase” part. https://aardvark.ucsd.edu/grad_conference/hagar.pdf https://aardvark.ucsd.edu/grad_conference/hagar.pdf In Gibbs’ approach one represents the microstate of a physical system with N particles each with f degrees of freedom by a point X ∈ Γ where Γ, the phase space of the system, is a 2Nf-dimensional space spanned by the Nf momenta and Nf configuration axes. As the system evolves this representative point will trace out a trajectory in Γ which obeys Hamilton’s equations of motion. Next, one considers a fictitious ensemble of individual systems (represented by a ‘cloud’, or a ‘fluid’, of points on phase space) each in a microstate compatible with a given macrostate (say, such and such energy in such and such pressure contained in such and such volume). The macroscopic parameters thus pick out a distribution of points in Γ. We then ascribe a normalized density function to the ensemble, ρ(p,q,t), and, except for entropy and temperature, the mean value of phase function with respect to ρ describes the system’s thermodynamic properties.[…] All this is so very fine, but if Gibbs’ systems obey Hamilton’s equations of motion then the ‘cloud’ representing them in phase space swarms like an incompressible fluid. Consequently his ‘fine-grained’ entropy as defined in (7.9) is invariant under the Hamiltonian flow: dSF/dt G(ρ) = 0. (7.10) If there is a problem in SFG it is not just the fact that it does not move. Recall that TD entropy is defined only in equilibrium, so in order to construct a mechanical counterpart one only needs to find a function whose value at a later equilibrium state is higher than at an earlier equilibrium state.21 But since the macroscopic parameters change between the two equilibrium states, the Gibbs’ approach has no problem in doing this just by defining a new ensemble with a new probability distribution for the new equilibrium state and this will match the thermodynamic entropy as before. […] it is not fair to use the macro-parameters, which are supposed to be derived from the micro-parameters, in order to construct the latter. In other words, the ensemble at later equilibrium state should be the Hamiltonian-time-evolved ensemble of the earlier equilibrium state, otherwise the system is not governed by Hamilton’s equations as one originally presupposes. Thus, if one wants to use Gibbs’ fine-grained entropy as a mechanical counterpart to TD entropy, then one must abandon standard, Hamiltonian, dynamics since it does not connect the two fine-grained equilibrium states. That this is the true problem with Gibbs’ fine-grained entropy escaped many commentators, and as a result the foundations of SM were soon piled with a lot of dead wood. Stemming from the famous Ehrenfests’ paper (1912, 43–79) where Boltzmann’s students complained on Gibbs’ treatment of irreversibility by categorizing it bluntly as “incorrect”, the last century was consumed with attempts to find a monotonically increasing function as a counterpart for TD entropy. One way to achieve this goal is to follow Gibbs himself, who introduces the mathematical trick of ‘coarse graining’ and devises new notions of entropy and equilibrium. In this approach one divides Γ into many small finite cells of volume ω and then takes the average of ρ over these cells.