# Liouville's theorem and time evolution of ensemble average

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• Jeremy1986
In summary, the conversation discusses Liouville's theorem and its application in calculating the time evolution of the ensemble average of a quantity. The integration by parts method is used to simplify the calculation, and the concept of total derivatives is clarified. The second question is about the difference between ##\frac{{d\rho }}{{dt}}## and ##\frac{{\partial \rho }}{{\partial t}}##, and it is explained that the latter only relates to the variation of a field with time at a fixed point in phase space. Overall, the conversation provides a better understanding of Liouville's theorem and its application in physics.
Jeremy1986
TL;DR Summary
Question about using Liouville's theorem to calculate time evolution of ensemble average
With the Liouville's theorem
$$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{d{q_a}}}{{dt}})} = 0$$
when we calculate the time evolution of the ensemble average of a quantity ## O(p,q)## we have
$$\frac{{d\left\langle O \right\rangle }}{{dt}} = \int {d\Gamma \frac{{\partial \rho (p,q,t)}}{{\partial t}}O(p,q)} = \sum\limits_{a = 1}^{3N} {\int {d\Gamma } } O(p,q)(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{\partial H}}{{\partial {q_a}}} - \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{\partial H}}{{\partial {p_a}}})$$
here ## p,q## represents a bunch of generalized coordinates and momentum ## {p_a},{q_a},a = 1,...,3N## .
Then by using the method of integration by parts, the above integration becomes
$$\frac{{d\left\langle O \right\rangle }}{{dt}} = - \sum\limits_{a = 1}^{3N} {\int {d\Gamma } } \rho [(\frac{{\partial O}}{{\partial {p_a}}}\frac{{\partial H}}{{\partial {q_a}}} - \frac{{\partial O}}{{\partial {q_a}}}\frac{{\partial H}}{{\partial {p_a}}}) + O(\frac{{{\partial ^2}H}}{{\partial {p_a}\partial {q_a}}} - \frac{{{\partial ^2}H}}{{\partial {q_a}\partial {p_a}}})$$Here comes my questions, I think the integration by parts uses ## \int {d{p_a}} \frac{{\partial \rho }}{{\partial {p_a}}} = \int {d\rho }## . However as
## \rho$~$\rho (p,q,t)## , should we have ## \frac{{d\rho }}{{d{p_a}}} = \frac{{\partial \rho }}{{\partial {p_a}}} + \frac{{\partial \rho }}{{\partial t}}\frac{{dt}}{{d{p_a}}}## .

I take that last relation for granted because when we calculate ## \frac{{dy}}{{dx}}## , if ## y=y(x)## determines implicitly by some relation ## F(x,y)=0## , we use
$$\frac{{\partial F(x,y)}}{{\partial x}} + \frac{{\partial F(x,y)}}{{\partial y}}\frac{{dy}}{{dx}} = 0$$ and get $$\frac{{dy}}{{dx}} = - \frac{{\frac{{\partial F(x,y)}}{{\partial x}}}}{{\frac{{\partial F(x,y)}}{{\partial y}}}}$$ If we differentiate ## F(x,y)=0## with y, in order to get the same value of ## \frac{{dy}}{{dx}}## , we need to have
$$\frac{{\partial F(x,y)}}{{\partial x}}\frac{{dx}}{{dy}} + \frac{{\partial F(x,y)}}{{\partial y}} = 0$$ where we consider ## x## ~## x(y)## . So back to the ## \rho## case, in calculating ## \frac{{d\rho }}{{d{p_a}}}## , I think ## \frac{{\partial \rho }}{{\partial t}}\frac{{dt}}{{d{p_a}}}## term should be taken into account since ## p_a## ~## p_a(t)## . Then the integration by parts seems to be wrong, so I think I have made a mistake.______________________________________________________________________

My second question concerns Liouville's theorem itself, if we have ## \frac{{d\rho }}{{dt}}=0## , then can we have ## \frac{{d\rho }}{{d{q_a}}} = \frac{{d\rho }}{{dt}}\frac{{d{q_a}}}{{dt}} = 0## ? It sounds ridiculous as it indicates that the probability density is everywhere the same in the phase space, regardless of what the system is. Then where did I make a mistake? Thanks for your patience reading my long questions!

Concerning the "integration by parts" question: In multidimensional integration "integration by parts" means to apply the generalized Gauss's theorem in arbitrary dimensions. E.g., for the first piece in your integral (Einstein summation convention applies!),
$$\int \mathrm{d} \Gamma O \frac{\partial \rho}{\partial p_a} \frac{\partial H}{\partial q_a} = \int \mathrm{d} \Gamma [\partial_{p_a} (O \rho \partial_{q_a} H)-\rho \partial_{p_a} (O \partial_{q_a} H)].$$
Now the first is a total divergence in ##p## space and the integral over ##\mathrm{d}^{3N} p## gives a hyper-surface integral due to Gauss's theorem. Since you assume that ##\rho## drops to 0 sufficiently fast at infinity of phase space, this contribution can be dropped, and you are left with the other phase-space integral. Combining everything leads to the desired result (just go on calculating; the result is very intuitive!).

The 2nd question doesn't make sense to me. What is a total derivative of ##\rho## wrt. ##q_a##? Of course ##\rho## is not uniform across phase space. This wouldn't be a properly normalized distribution: Integrating over entire phase space should give the particle number, which is finite!

Jeremy1986
vanhees71 said:
Concerning the "integration by parts" question: In multidimensional integration "integration by parts" means to apply the generalized Gauss's theorem in arbitrary dimensions. E.g., for the first piece in your integral (Einstein summation convention applies!),
$$\int \mathrm{d} \Gamma O \frac{\partial \rho}{\partial p_a} \frac{\partial H}{\partial q_a} = \int \mathrm{d} \Gamma [\partial_{p_a} (O \rho \partial_{q_a} H)-\rho \partial_{p_a} (O \partial_{q_a} H)].$$
Now the first is a total divergence in ##p## space and the integral over ##\mathrm{d}^{3N} p## gives a hyper-surface integral due to Gauss's theorem. Since you assume that ##\rho## drops to 0 sufficiently fast at infinity of phase space, this contribution can be dropped, and you are left with the other phase-space integral. Combining everything leads to the desired result (just go on calculating; the result is very intuitive!).

The 2nd question doesn't make sense to me. What is a total derivative of ##\rho## wrt. ##q_a##? Of course ##\rho## is not uniform across phase space. This wouldn't be a properly normalized distribution: Integrating over entire phase space should give the particle number, which is finite!
Thank you very much for your nice reply! I think I start to understand my first question. What I was puzzled is actually the difference between ##\frac{{d\rho }}{{dt}}## and ##\frac{{\partial \rho }}{{\partial t}}##, which I now realized defines change in ##\rho## along the path in phase space determined by Hamiltonian and defines the variation of a field "##\rho##" with time at a fixed point in phase space. For the case of ensemble average, the integration is just over phase space, it does not relates to any evolution of ##{q,p}## in the phase space, so the ##{q,p}## in the integration has nothing to do with time t. The only time dependence in ##\left\langle O \right\rangle## is the variation of ##\rho## at each point in phase space.

vanhees71

## 1. What is Liouville's theorem and how does it relate to time evolution of ensemble average?

Liouville's theorem is a fundamental principle in classical mechanics that states that the phase space density of an isolated system remains constant over time. This means that the distribution of positions and momenta of particles in a system will not change as the system evolves. This theorem is often used to describe the time evolution of an ensemble average, which is the average behavior of a large number of systems with similar initial conditions.

## 2. How is Liouville's theorem different from the conservation of energy?

While both Liouville's theorem and the conservation of energy deal with the behavior of systems over time, they are fundamentally different concepts. Liouville's theorem is a statement about the constancy of phase space density, while the conservation of energy is a statement about the constancy of the total energy in a closed system.

## 3. Can Liouville's theorem be applied to quantum systems?

No, Liouville's theorem only applies to classical systems and cannot be extended to quantum systems. However, there is a similar principle in quantum mechanics called the von Neumann equation, which describes the time evolution of the density matrix of a quantum system.

## 4. How does Liouville's theorem relate to the concept of entropy?

Liouville's theorem is closely related to the concept of entropy, which is a measure of the disorder or randomness of a system. According to Liouville's theorem, the phase space density of an isolated system remains constant, which means that the system's entropy will also remain constant over time.

## 5. Are there any limitations to Liouville's theorem?

Yes, Liouville's theorem only applies to isolated systems with no external influences. In real-world situations, it is often difficult to find truly isolated systems, and external forces can cause changes to the system's phase space density. In these cases, Liouville's theorem may not accurately describe the time evolution of the system.

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