i*****e
发帖数: 218 | 1 像大家请教一个问题：
热力学第二定律（熵增原理）在统计力学中是否可以从更基本的原理推导出来 ？
还是在统计力学中， 熵增原理仍然当作一个基本定律， 没法从更基本的原理推导出来。
翻了手头的基本统计力学的书， 没发现推导热力学第二定律（熵增原理）的。
如果能推导出来， 大家能否给一个推导它的链接 ？ | j**p
发帖数: 53 | 2 1. Boltzmann's H-theorem is not really a proof of the second law because of
the "molecular chaos" assumtion it relies on.
2. One way to understand the second law is: entropy increases due to coarse-
graining in measurement. More details is as follows:
consider a 2-state system. Its entropy is (denoting its probability in the
lower state as p) by definition
S = -pln(p)-(1-p)ln(1-p)
Let's say p is very close to 50%, p = (1/2) + a with a<<1. Then by doing a
taylor expansion of the above entropy definition, you can see that
S_true = ln(2) + O(a) < ln(2).
However, when we measure probability p in reality, due to measurement error,
we may not be able to tell (1/2) + a from (1/2) - a (e.g. 0.5+1.0e-32 and
0.5-1.0e-32 all come out as 0.5 in our experiment), and think p = 1/2. Then
measured entroy is S_measured = ln(2) > S_true.
Coarse-graining in measurment causes entropy to "increase" as shown above.
Read K Thorne's book for example for a more detailed exposition of the above. | j**p
发帖数: 53 | 3 for expositions of this topic with more mathematical rigor, please see
Theorem 11.9 of "Quantum Computation and Quantum Information" By Michael A.
Nielsen, Isaac L. Chuang
and
http://en.wikipedia.org/wiki/Quantum_relative_entropy | i*****e
发帖数: 218 | 4 》Coarse-graining in measurment causes entropy to "increase"
你是说， 熵增原理是由于“测量的”不够细致 ？
of
coarse-
【在 j**p 的大作中提到】
: 1. Boltzmann's H-theorem is not really a proof of the second law because of
: the "molecular chaos" assumtion it relies on.
: 2. One way to understand the second law is: entropy increases due to coarse-
: graining in measurement. More details is as follows:
: consider a 2-state system. Its entropy is (denoting its probability in the
: lower state as p) by definition
: S = -pln(p)-(1-p)ln(1-p)
: Let's say p is very close to 50%, p = (1/2) + a with a<<1. Then by doing a
: taylor expansion of the above entropy definition, you can see that
: S_true = ln(2) + O(a) < ln(2).
| j**p
发帖数: 53 | |
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