Boltzmann I Never Knew Ye'
Well apparently there is quite a bit which I either failed to pick up on during Thermo last year or lots of stuff was glossed over.
The entropy I knew and loved was simply the logarithm of the multiplicity of a state, or (if an ideal gas) some derivatives of internal energy and temperature. A closed system's entropy tends to increase -the system evolves towards equilibrium- because all microstates are equally accessable. These rules are all talked about in terms of simple isolated dankenexperiments like einstein solids (bricks) or gas filled pistons.
Now apparently, after figuring all of this out -which was where I left the story-, Boltzmann went on to worry about why adding probability + mechanics removes detirmanism. eg Gasses are just molecules running into each other, this is a reversable process, so why cant I get the air back in my tire? In his paper, "Cosmic Inflation and the Arrow of Time", Andreas Albrecht writes,
Well, Boltzmann didnt teach ME this.
So first I google "boltzmann low entropy initial conditions". From Soshichi Uchii at Kyoto U I have my first hit . She has an essay on reductionism (one theory "reduces" to another) with thermo as the example. She talks about Boltzmann's fight to reconcile mechanics with probability and the irreversability probability adds, finally ending with his "Ergodic Hypothesis". She is a little unclear about its other meanings besides all microstates are equally accessable. According to Laura Cupple's old website at Davidson, another reading of the hypothesis is:
Thus Boltzmann defines probability in terms of dynamical properties in an attempt to justify additional probabilistic assumptions.
This short essay by Astri Kleppe in Norway is also very good. However still noone is mentioning initial conditions. Well so much for the google quick fix. I have walked ALL the way down to the computer lab from office to print,
The entropy I knew and loved was simply the logarithm of the multiplicity of a state, or (if an ideal gas) some derivatives of internal energy and temperature. A closed system's entropy tends to increase -the system evolves towards equilibrium- because all microstates are equally accessable. These rules are all talked about in terms of simple isolated dankenexperiments like einstein solids (bricks) or gas filled pistons.
Now apparently, after figuring all of this out -which was where I left the story-, Boltzmann went on to worry about why adding probability + mechanics removes detirmanism. eg Gasses are just molecules running into each other, this is a reversable process, so why cant I get the air back in my tire? In his paper, "Cosmic Inflation and the Arrow of Time", Andreas Albrecht writes,
...Boltzmann taught us that the thermodynamic arrow of time
arises from very non-generic ("low entropy") initial conditions.
arises from very non-generic ("low entropy") initial conditions.
Well, Boltzmann didnt teach ME this.
So first I google "boltzmann low entropy initial conditions". From Soshichi Uchii at Kyoto U I have my first hit . She has an essay on reductionism (one theory "reduces" to another) with thermo as the example. She talks about Boltzmann's fight to reconcile mechanics with probability and the irreversability probability adds, finally ending with his "Ergodic Hypothesis". She is a little unclear about its other meanings besides all microstates are equally accessable. According to Laura Cupple's old website at Davidson, another reading of the hypothesis is:
The Ergodic Hypothesis states that the dynamical probability of finding a physical system in a particular state (X) is
T(X)/T = Omega(X)/Omega
T(X) is the fraction of time (out of total time T) spent in state X.
Omega(X) is X's fraction of the number of states (out of the total number Omega).
T(X)/T = Omega(X)/Omega
T(X) is the fraction of time (out of total time T) spent in state X.
Omega(X) is X's fraction of the number of states (out of the total number Omega).
Thus Boltzmann defines probability in terms of dynamical properties in an attempt to justify additional probabilistic assumptions.
This short essay by Astri Kleppe in Norway is also very good. However still noone is mentioning initial conditions. Well so much for the google quick fix. I have walked ALL the way down to the computer lab from office to print,
- "Cosmology, Time's Arrow and That Old Double Standard", Huw Price 1994
- "Cosmic Inflation and the Arrow of Time", Andreas Albrecht
These are references 1 and 2 from Carroll's "Arrow". Maybe I will feel less indignant after reading more than Albrecht's abstract.
Labels: Entropy, Readings, Time's Arrow
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