热力学量 U6gE S=S(E,N,V)=kB In(E,N,V) Boltzmann关系 d In=ds/kB=BdE Bpdv -BudN Gibbs熵 S=-ksTr{lnp}=-ka∑(s)=-kB∑PssIPss =-ks∑P:InPs Shannon嫡：ps处在s态上的几率 (slE.E)
热力学量 𝑈 = 𝑇𝑟{H 𝜌ˆ} = 1 Ω 𝑇𝑟{H𝛿𝐸,H} = 1 Ω 𝐸Ω = 𝐸 𝑆 = 𝑆(𝐸, 𝑁, 𝑉) = 𝑘𝐵 ln Ω(𝐸, 𝑁, 𝑉) ✞ ✝ ☎ ✆ Boltzmann 关系 𝑑𝑆 = 𝑑𝐸 𝑇 + 𝑝 𝑇 𝑑𝑉 − 𝜇 𝑇 𝑑𝑁 𝑑 ln Ω = 𝑑𝑆/𝑘𝐵 = 𝛽𝑑𝐸 + 𝛽𝑝𝑑𝑉 − 𝛽𝜇𝑑𝑁 1 𝑇 = ( 𝜕𝑆 𝜕𝐸 ) 𝑁𝑉 𝑝 𝑇 = ( 𝜕𝑆 𝜕𝑉 ) 𝐸 𝑁 − 𝜇 𝑇 = ( 𝜕𝑆 𝜕𝑁 ) 𝐸𝑉 Gibbs 熵 𝑆 = −𝑘𝐵𝑇𝑟{𝜌ˆ ln ˆ𝜌} = −𝑘𝐵 ∑ 𝑠 h𝑠|𝜌ˆ ln ˆ𝜌|𝑠i = −𝑘𝐵 ∑ 𝑠 𝜌𝑠𝑠 ln 𝜌𝑠𝑠 = −𝑘𝐵 ∑ 𝑠 𝑝𝑠 ln 𝑝𝑠 ✞ ✝ ☎ ✆ Shannon 熵：𝑝𝑠 处在 𝑠 态上的几率 = −𝑘𝐵 ∑ {𝑠|𝐸𝑠=𝐸} 1 Ω ln 1 Ω = 𝑘𝐵 ∑ {𝑠|𝐸𝑠=𝐸} 1 Ω ln Ω = 𝑘𝐵 ln Ω
各种各样的熵 o.Boltzmann熵：S=kBln2 0 Gibbs熵、von Neumann嫡：S=-kBTr{plnp Q Shannon熵/信息熵：S=-∑s Ps In ps Shannon熵和Gibbs熵基本相同，相差一个kB Renyi嫡：Sa=己InTr{}“=己aln∑sp Sa-1=Shannon熵 Tsallis熵：Sg=己gTr{p9}-刂 矿Sg=l=Shannon熵
各种各样的熵 Boltzmann 熵：𝑆 = 𝑘𝐵 ln Ω Gibbs 熵、von Neumann 熵：𝑆 = −𝑘𝐵𝑇𝑟{𝜌ˆ ln ˆ𝜌} Shannon 熵／信息熵：𝑆 = − ∑ 𝑠 𝑝𝑠 ln 𝑝𝑠 ☞Shannon 熵和 Gibbs 熵基本相同，相差一个 𝑘𝐵 Rényi 熵：𝑆𝛼 = 1 1−𝛼 ln𝑇𝑟{𝜌ˆ} 𝛼 = 1 1−𝛼 ln ∑ 𝑠 𝑝 𝛼 𝑠 ☞𝑆𝛼=1 =Shannon 熵 Tsallis 熵：𝑆𝑞 = 1 1−𝑞 [𝑇𝑟{𝜌 𝑞 } − 1] ☞𝑆𝑞=1 =Shannon 熵
Maxwell's Demon Maxwell设想的一种小生物(finite being),可以把快慢分 子原子分开来，这样可以自发的创造温度差，违背热力学第 二定律。 o.Kelvin把这个“生物”称为Maxwel's demon The defination of a"demon"according to the use of this word by Maxwell,is an intelligent being endowed with free will,and fine enough tactile and perceptive organsation to give him the faculty of observing and influencing individual molecules of matter
Maxwell’s Demon Maxwell 设想的一种小生物（finite being）, 可以把快慢分 子/原子分开来，这样可以自发的创造温度差，违背热力学第 二定律。 Kelvin 把这个“生物”称为 Maxwell’s demon The defination of a “demon” according to the use of this word by Maxwell, is an intelligent being endowed with free will, and fine enough tactile and perceptive organsation to give him the faculty of observing and influencing individual molecules of matter
"The Sorting Demon of Maxwell",Kelvin,Nature 20,126 (1879) THE word "demon"which originally in Greek meant a supernatural being,has never been properly used to signify a real or ideal personifi- cation of malignity. Clerk Maxwell's "demon"is a creature of imagination having cer- tain perfectly well-defined powers of action,purely mechanical in their character,invented to help us fo understand the "Dissipation of En- ergy”in nature. He is a being with no preternatural qualities,and differs from real living animals only in extreme smallness and agility.He can at pleasure stop, or strike,or push,or pull any single atom of matter,and so moderate its natural course of motion.Endowed ideally with arms and hands and fingers-two hands and ten fingers suffice-he can do as much for atoms as a pianoforte player can do for the keys of the piano-just a little more,he can push or pull each atom in any direction
“The Sorting Demon of Maxwell”, Kelvin, Nature 20, 126 (1879) THE word “demon”which originally in Greek meant a supernatural being, has never been properly used to signify a real or ideal personification of malignity. Clerk Maxwell’s “demon”is a creature of imagination having certain perfectly well-defined powers of action, purely mechanical in their character, invented to help us fo understand the “Dissipation of Energy”in nature. He is a being with no preternatural qualities, and differs from real living animals only in extreme smallness and agility. He can at pleasure stop, or strike, or push, or pull any single atom of matter, and so moderate its natural course of motion. Endowed ideally with arms and hands and fingers—two hands and ten fingers suffice —he can do as much for atoms as a pianoforte player can do for the keys of the piano—just a little more, he can push or pull each atom in any direction
Thermodynamic Exorcism Maxwell本人并不认为这是个大问题，只是揭示了如果能够 在原子/分子层面操控系统的话，就可以规避热力学第二定 律。 o.Smoluchowski等人(1912年)认为如果这个demon也服从物 理定律的话，那么要么热涨落会让demon失效，要么demon 运作过程需要消耗能量，导致整个宇宙嫡。 o Szilard(1929年)认为获得/测量分子运动速度信息需要消 耗能量。这些工作首次揭示了信息处理与熵的关系。 o.von Neumann(1949年)和Brillouin(1962年)认为处理一 个bit的信息需要消耗kBT In2的能量。 0 Landauer(1961年)发现可逆的测量/获得信息不需要消耗 能量，但是删除一个bit的信息需要消耗ksTn2的能量。 Q信息熵和热力学熵没有区别，可以通过消耗信息熵来做功
Thermodynamic Exorcism Maxwell 本人并不认为这是个大问题，只是揭示了如果能够 在原子／分子层面操控系统的话，就可以规避热力学第二定 律。 Smoluchowski 等人（1912 年）认为如果这个 demon 也服从物 理定律的话，那么要么热涨落会让 demon 失效，要么 demon 运作过程需要消耗能量，导致整个宇宙熵。 Szilard（1929 年）认为获得／测量分子运动速度信息需要消 耗能量。这些工作首次揭示了信息处理与熵的关系。 von Neumann （1949 年）和 Brillouin（1962 年）认为处理一 个 bit 的信息需要消耗 𝑘𝐵𝑇 ln 2 的能量。 Landauer（1961 年）发现可逆的测量／获得信息不需要消耗 能量，但是删除一个 bit 的信息需要消耗 𝑘𝐵𝑇 ln 2 的能量。 信息熵和热力学熵没有区别，可以通过消耗信息熵来做功