巨觀世界之穩定性，熱力學的有效性，促使玻爾茲曼運用物理運動定律、統計方法，思考巨量微觀粒子之構成系統，企圖從中推導出整體熱力學。這個歷史進程中，有件困擾玻爾茲曼的事在於對稱性的破壞︰怎麼可能每個粒子都有時間對稱性，那個構成系統的整體卻沒有的呢？

展示時間對稱性蹺蹺板
它最後將導向何方？？

物理系統的『時間對稱性』T-symmetry，是說『物理定律』在『時間反向變換』time reversal transformation $T: t \mapsto -t$ 下保持不變。比方說『牛頓第二運動定律』 $\vec{F} = m \frac{d}{d[-t]} \frac{d}{d[-t]} \vec{r} = m \frac{d}{dt} \frac{d}{dt} \vec{r}$ 具有『時間對稱性』。假使一個粒子從『初始態』 $(\vec{r_i} , \vec{p_i})$ 沿著軌跡往『終止態』 $(\vec{r_f} , \vec{p_f})$ 運動，如果『時間逆流』，此粒子將逆向由『終止態』 $(\vec{r_f} , \vec{p_f})$ 沿著軌跡向『初始態』 $(\vec{r_i} , \vec{p_i})$ 運動。

於是在『理化系統』中，就有了『微觀可逆性』原理︰

Corresponding to every individual process there is a reverse process, and in a state of equilibrium the average rate of every process is equal to the average rate of its reverse process.

。然而這個『微觀可逆性』原理，到了『巨觀世界』後，卻是與『熱力學』的『最大熵』 Maximum entropy 理論衝突。一八七二年時，玻爾茲曼提出了『 H 理論』︰

$H(t) = \int \limits_0^{\infty} f(E,t) \left[ \log\left(\frac{f(E,t)}{\sqrt{E}}\right) - 1 \right] \, dE$ ，此處 $f(E,t)$ 就是在 $t$ 時間的『能量分布』函數，而那個 $f(E,t) dE$ 是『動能』在 $E$ 與 $E+dE$ 間之『粒子數』。據聞，玻爾茲曼是想用著『統計力學』的辦法，能夠推導出『最大熵』 $S$ 的『不可逆性』。

馬克士威妖

可以如是描述成︰假使一個絕熱容器被分成兩塊，中間有『妖』所控制之『門』，那個容器中的『粒子』到處亂撞時，總會碰到『門』上，此『妖』喜歡將『快‧慢』之『粒子』分別為『兩半』，因此，其中的一半就會比另外一半的『溫度』要高。

由於更早五年前，『馬克士威』設想了一個『想像實驗』︰

… if we conceive of a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, whose attributes are as essentially finite as our own, would be able to do what is impossible to us. For we have seen that molecules in a vessel full of air at uniform temperature are moving with velocities by no means uniform, though the mean velocity of any great number of them, arbitrarily selected, is almost exactly uniform. Now let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower molecules to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics.

也就有人『Johann Loschmidt』反對玻爾茲曼的『 H 理論』之說法︰if there is a motion of a system from time t0 to time t1 to time t2 that leads to a steady decrease of H (increase of entropy) with time, then there is another allowed state of motion of the system at t1, found by reversing all the velocities, in which H must increase. This revealed that one of Boltzmann’s key assumptions, molecular chaos, or, the Stosszahlansatz, that all particle velocities were completely uncorrelated, did not follow from Newtonian dynamics.

─── 摘自《物理哲學·下》

難到『詳細平衡』之基本原理有錯誤？？

Detailed balance

The principle of detailed balance is formulated for kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions): At equilibrium, each elementary process should be equilibrated by its reverse process.

History

The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle.^[1] The arguments in favor of this property are founded upon microscopic reversibility.^[2]

Five years before Boltzmann, James Clerk Maxwell used the principle of detailed balance for gas kinetics with the reference to the principle of sufficient reason.^[3] He compared the idea of detailed balance with other types of balancing (like cyclic balance) and found that “Now it is impossible to assign a reason” why detailed balance should be rejected (pg. 64).

Albert Einstein in 1916 used the principle of detailed balance in a background for his quantum theory of emission and absorption of radiation.^[4]

In 1901, Rudolf Wegscheider introduced the principle of detailed balance for chemical kinetics.^[5] In particular, he demonstrated that the irreversible cycles ${\ce {A1->A2->\cdots ->A_{\mathit {n}}->A1}}$ are impossible and found explicitly the relations between kinetic constants that follow from the principle of detailed balance. In 1931, Lars Onsager used these relations in his works,^[6] for which he was awarded the 1968 Nobel Prize in Chemistry.

The principle of detailed balance has been used in Markov chain Monte Carlo methods since their invention in 1953.^[7]^[8] In particular, in the Metropolis–Hastings algorithm and in its important particular case, Gibbs sampling, it is used as a simple and reliable condition to provide the desirable equilibrium state.

Now, the principle of detailed balance is a standard part of the university courses in statistical mechanics, physical chemistry, chemical and physical kinetics.^[9]^[10]^[11]

Microscopical background

The microscopic “reversing of time” turns at the kinetic level into the “reversing of arrows”: the elementary processes transform into their reverse processes. For example, the reaction

\sum _{i}\alpha _{i}{\ce {A}}_{i}{\ce {->}}\sum _{j}\beta _{j}{\ce {B}}_{j}

transforms into

\sum _{j}\beta _{j}{\ce {B}}_{j}{\ce {->}}\sum _{i}\alpha _{i}{\ce {A}}_{i}

and conversely. (Here, ${\ce {A}}_{i},{\ce {B}}_{j}$ are symbols of components or states, $\alpha _{i},\beta _{j}\geq 0$ are coefficients). The equilibrium ensemble should be invariant with respect to this transformation because of microreversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process.

This reasoning is based on three assumptions:

${\ce {A}}_{i}$ does not change under time reversal;
Equilibrium is invariant under time reversal;
The macroscopic elementary processes are microscopically distinguishable. That is, they represent disjoint sets of microscopic events.

Any of these assumptions may be violated.^[12] For example, Boltzmann’s collision can be represented as ${\ce {{A_{\mathit {v}}}+A_{\mathit {w}}->{A_{\mathit {v'}}}+A_{\mathit {w'}},}}$ where ${\ce {A}}_{v}$ is a particle with velocity v. Under time reversal ${\ce {A}}_{v}$ transforms into ${\ce {A}}_{-v}$ . Therefore, the collision is transformed into the reverse collision by the PT transformation, where P is the space inversion and T is the time reversal. Detailed balance for Boltzmann’s equation requires PT-invariance of collisions’ dynamics, not just T-invariance. Indeed, after the time reversal the collision ${\ce {{A_{\mathit {v}}}+A_{\mathit {w}}->{A_{\mathit {v'}}}+A_{\mathit {w'}},}}$ transforms into ${\ce {{A_{\mathit {-v'}}}+A_{\mathit {-w'}}->{A_{\mathit {-v}}}+A_{\mathit {-w}}.}}$ For the detailed balance we need transformation into ${\ce {{A_{\mathit {v'}}}+A_{\mathit {w'}}->{A_{\mathit {v}}}+A_{\mathit {w}}.}}$ For this purpose, we need to apply additionally the space reversal P. Therefore, for the detailed balance in Boltzmann’s equation not T-invariance but PT-invariance is needed.

Equilibrium may be not T– or PT-invariant even if the laws of motion are invariant. This non-invariance may be caused by the spontaneous symmetry breaking. There exist nonreciprocal media (for example, some bi-isotropic materials) without T and PT invariance.^[12]

If different macroscopic processes are sampled from the same elementary microscopic events then macroscopic detailed balance^{[clarification needed]} may be violated even when microscopic detailed balance holds.^[12]^[13]

Now, after almost 150 years of development, the scope of validity and the violations of detailed balance in kinetics seem to be clear.

……

Detailed balance and entropy increase

For many systems of physical and chemical kinetics, detailed balance provides sufficient conditions for the strict increase of entropy in isolated systems. For example, the famous Boltzmann H-theorem^[1] states that, according to the Boltzmann equation, the principle of detailed balance implies positivity of entropy production. The Boltzmann formula (1872) for entropy production in rarefied gas kinetics with detailed balance^[1]^[2] served as a prototype of many similar formulas for dissipation in mass action kinetics^[15] and generalized mass action kinetics^[16] with detailed balance.

Nevertheless, the principle of detailed balance is not necessary for entropy growth. For example, in the linear irreversible cycle ${\ce {A1->A2->A3->A1}}$ , entropy production is positive but the principle of detailed balance does not hold.

Thus, the principle of detailed balance is a sufficient but not necessary condition for entropy increase in Boltzmann kinetics. These relations between the principle of detailed balance and the second law of thermodynamics were clarified in 1887 when Hendrik Lorentz objected to the Boltzmann H-theorem for polyatomic gases.^[17] Lorentz stated that the principle of detailed balance is not applicable to collisions of polyatomic molecules.

Boltzmann immediately invented a new, more general condition sufficient for entropy growth.^[18] Boltzmann’s condition holds for all Markov processes, irrespective of time-reversibility. Later, entropy increase was proved for all Markov processes by a direct method.^[19]^[20] These theorems may be considered as simplifications of the Boltzmann result. Later, this condition was referred to as the “cyclic balance” condition (because it holds for irreversible cycles) or the “semi-detailed balance” or the “complex balance”. In 1981, Carlo Cercignani and Maria Lampis proved that the Lorentz arguments were wrong and the principle of detailed balance is valid for polyatomic molecules.^[21] Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.

───

也許該說的是詳細平衡原理有假設前提。自然界裡事實存在著反例！就像也有

自發對稱破缺

自發對稱破缺（spontaneous symmetry breaking）是某些物理系統實現對稱性破缺的模式。當物理系統所遵守的自然定律具有某種對稱性，而物理系統本身並不具有這種對稱性，則稱此現象為自發對稱破缺。^[1]^:141^[2]^:125這是一種自發性過程（spontaneous process），由於這過程，本來具有這種對稱性的物理系統，最終變得不再具有這種對稱性，或不再表現出這種對稱性，因此這種對稱性被隱藏。因為自發對稱破缺，有些物理系統的運動方程式或拉格朗日量遵守這種對稱性，但是最低能量解答不具有這種對稱性。從描述物理現象的拉格朗日量或運動方程式，可以對於這現象做分析研究。

對稱性破缺主要分為自發對稱破缺與明顯對稱性破缺兩種。假若在物理系統的拉格朗日量裏存在著一個或多個違反某種對稱性的項目，因此導致系統的物理行為不具備這種對稱性，則稱此為明顯對稱性破缺。

如右圖所示，假設在墨西哥帽（sombrero）的帽頂有一個圓球。這個圓球是處於旋轉對稱性狀態，對於繞著帽子中心軸的旋轉，圓球的位置不變。這圓球也處於局部最大重力勢的狀態，極不穩定，稍加微擾，就可以促使圓球滾落至帽子谷底的任意位置，因此降低至最小重力勢位置，使得旋轉對稱性被打破。儘管這圓球在帽子谷底的所有可能位置因旋轉對稱性而相互關聯，圓球實際實現的帽子谷底位置不具有旋轉對稱性──對於繞著帽子中心軸的旋轉，圓球的位置會改變。^[3]^:203

大多數物質的簡單相態或相變，例如晶體、磁鐵、一般超導體等等，可以從自發對稱破缺的觀點來了解。像分數量子霍爾效應（fractional quantum Hall effect）一類的拓撲相（topological phase）物質是值得注意的例外。^[4]

墨西哥帽位能函數的電腦繪圖，對於繞著帽子中心軸的旋轉，帽頂具有旋轉對稱性，帽子谷底的任意位置不具有旋轉對稱性，在帽子谷底的任意位置會出現對稱性破缺。

的事一樣！！如是能否理解波動‧耗散定理耶☆

Fluctuation-dissipation theorem

The fluctuation-dissipation theorem (FDT) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermal fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation-dissipation theorem applies both to classical and quantum mechanical systems.

The fluctuation-dissipation theorem relies on the assumption that the response of a system in thermodynamic equilibrium to a small applied force is the same as its response to a spontaneous fluctuation. Therefore, the theorem connects the linear response relaxation of a system from a prepared non-equilibrium state to its statistical fluctuation properties in equilibrium.^[1] Often the linear response takes the form of one or more exponential decays.

The fluctuation-dissipation theorem was originally formulated by Harry Nyquist in 1928,^[2] and later proven by Herbert Callen and Theodore A. Welton in 1951.^[3]

Qualitative overview and examples

The fluctuation-dissipation theorem says that when there is a process that dissipates energy, turning it into heat (e.g., friction), there is a reverse process related to thermal fluctuations. This is best understood by considering some examples:

Drag and Brownian motion

If an object is moving through a fluid, it experiences drag (air resistance or fluid resistance). Drag dissipates kinetic energy, turning it into heat. The corresponding fluctuation is Brownian motion. An object in a fluid does not sit still, but rather moves around with a small and rapidly-changing velocity, as molecules in the fluid bump into it. Brownian motion converts heat energy into kinetic energy—the reverse of drag.

Resistance and Johnson noise

If electric current is running through a wire loop with a resistor in it, the current will rapidly go to zero because of the resistance. Resistance dissipates electrical energy, turning it into heat (Joule heating). The corresponding fluctuation is Johnson noise. A wire loop with a resistor in it does not actually have zero current, it has a small and rapidly-fluctuating current caused by the thermal fluctuations of the electrons and atoms in the resistor. Johnson noise converts heat energy into electrical energy—the reverse of resistance.

Light absorption and thermal radiation

When light impinges on an object, some fraction of the light is absorbed, making the object hotter. In this way, light absorption turns light energy into heat. The corresponding fluctuation is thermal radiation (e.g., the glow of a “red hot” object). Thermal radiation turns heat energy into light energy—the reverse of light absorption. Indeed, Kirchhoff’s law of thermal radiation confirms that the more effectively an object absorbs light, the more thermal radiation it emits.

Examples in detail

The fluctuation-dissipation theorem is a general result of statistical thermodynamics that quantifies the relation between the fluctuations in a system at thermal equilibrium and the response of the system to applied perturbations.

The model thus allows, for example, the use of molecular models to predict material properties in the context of linear response theory. The theorem assumes that applied perturbations, e.g., mechanical forces or electric fields, are weak enough that rates of relaxation remain unchanged.

Brownian motion

For example, Albert Einstein noted in his 1905 paper on Brownian motion that the same random forces that cause the erratic motion of a particle in Brownian motion would also cause drag if the particle were pulled through the fluid. In other words, the fluctuation of the particle at rest has the same origin as the dissipative frictional force one must do work against, if one tries to perturb the system in a particular direction.

From this observation Einstein was able to use statistical mechanics to derive the Einstein-Smoluchowski relation

D = {\mu \, k_B T}

which connects the diffusion constant D and the particle mobility μ, the ratio of the particle’s terminal drift velocity to an applied force. k_B is the Boltzmann constant, and T is the absolute temperature.

Thermal noise in a resistor

In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance R, $k_BT$ , and the bandwidth $\Delta\nu$ over which the voltage is measured:

\langle V^2 \rangle = 4Rk_BT\,\Delta\nu.

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每日彙整: 2017-02-20

時間序列︰當波動遇上耗散