# 物理代写|宇宙学代写cosmology代考|PHYS3080

## 物理代写|宇宙学代写cosmology代考|Boltzmann equation

After having treated gravity in the homogeneous universe, let us now turn to the equations governing matter and radiation. In cosmology, we are not interested in the fate of individual particles, but in their behavior in a statistical sense. Hence let us consider a collection of particles occupying some region of space, as we did in Sect. 2.3. In classical physics, these particles are completely described by the set $\left{\boldsymbol{x}_i, \boldsymbol{p}_i\right}$ of their positions $\boldsymbol{x}_i$ and momenta $\boldsymbol{p}_i$. We can then define the distribution function, as in Sect. 2.3, by relating it to the number of particles in a small phase-space element around $(\boldsymbol{x}, \boldsymbol{p})$ :
$$N(\boldsymbol{x}, \boldsymbol{p}, t)=f(\boldsymbol{x}, \boldsymbol{p}, t)(\Delta x)^3 \frac{(\Delta p)^3}{(2 \pi)^3} .$$
In the limit of a large number of particles within the volume element considered, $f(\boldsymbol{x}, \boldsymbol{p}, t)$ approaches a continuous function describing the state of the collection of particles, and we no longer need to keep track of individual particles. We already saw that the appropriate integration measure (in natural units) is given by $d^3 x d^3 p /(2 \pi)^3$. Note that we do not need to include the energy as a separate variable, since, at any point in phase space, $E$ is completely determined by $(x, p)$.

Now we would like to derive an equation governing this distribution function. This equation should uniquely follow from the equations of motion obeyed by the individual particles. Let us begin by neglecting any particle-particle interactions. Then, the only forces acting on the particles are long-range forces, which we can describe through a force field (more precisely, acceleration field) $a(x, p, t)$. This could for example be gravity, in which case $a=-\nabla \Psi(x, t)$, where the gravitational potential $\Psi$ (defined in Eq. (3.49) below) is independent of the particle momenta, or it could be the Lorentz force due to electromagnetic fields. Then, using the definition of the momentum $p$, the equations of motion for nonrelativistic particles are
$$\dot{x}=\frac{p}{m} ; \quad \dot{p}=m a(x, p, t) .$$
The number of particles is conserved, which we can formalize by stating that the total time derivative of $f$ vanishes,
$$\frac{d f(\boldsymbol{x}, \boldsymbol{p}, t)}{d t}=0 \quad \text { where } \quad \frac{d}{d t}=\frac{\partial}{\partial t}+\dot{\boldsymbol{x}} \cdot \nabla_x+\dot{\boldsymbol{p}} \cdot \nabla_p$$
is the total (rather than partial) time derivative, and $\nabla_x, \nabla_p$ denote the gradient with respect to the arguments $x$ and $p$, respectively.

## 物理代写|宇宙学代写cosmology代考|Boltzmann equation for particles in a harmonic potential

Let us begin our journey with the Boltzmann equation with the case of nonrelativistic particles governed by a simple $x^2$ potential in one dimension. This Boltzmann equation exhibits all essential features of the full general-relativistic versions of the Boltzmann equation we will encounter in the next section, but the algebra is much less cumbersome. So here the physics will be quite transparent. It will be useful to keep this example in mind when the algebra threatens to obscure the physics in the following chapters.

Consider free particles living in a one-dimensional harmonic potential well. Their energy then is simply
$$E=\frac{p^2}{2 m}+\frac{1}{2} k x^2,$$
where $k$ is the spring constant. The distribution function is now a function of three scalar arguments $f=f(x, \beta ; t)$ : Fig: 3 : $^2$ illustrates the movement though phase space of a distribution of such particles (throughout, we consider the collisionless case $C[f]=0$ ). The full time derivative $d f / d t$ vanishes since the number of particles in the bunch at $t_1$ equals that at $t_2$. What changes over time is the location of the particles in phase space themselves. Alternatively, we can think of $x$ and $p$ as independent variables (not dependent on $t$ ) and take partial derivatives of $f$ with respect to $t, x$, and $p$. All of these partial derivatives are nonzero, but the appropriate weighted sum of the three vanishes [Eq. (3.17)].

To determine the coefficients $\dot{x}$ and $\dot{p}$ in Eq. (3.17), we must use the equations of motion, i.e. the one-dimensional version of Eq. (3.16). Via Newton’s force law, we have
$$\dot{x}=\frac{p}{m} \quad \text { and } \quad \dot{p}=-k x .$$
When generalizing to the relativistic case, these familiar equations will be replaced by the geodesic equation we have derived in Sect. 2.1.2. The collisionless Boltzmann equation for the present case is then
$$\frac{\partial f}{\partial t}+\frac{p}{m} \frac{\partial f}{\partial x}-k x \frac{\partial f}{\partial p}=0 .$$

## 物理代写|宇宙学代写cosmology代考|玻尔兹曼方程

$$N(\boldsymbol{x}, \boldsymbol{p}, t)=f(\boldsymbol{x}, \boldsymbol{p}, t)(\Delta x)^3 \frac{(\Delta p)^3}{(2 \pi)^3} .$$

$$\dot{x}=\frac{p}{m} ; \quad \dot{p}=m a(x, p, t) .$$

$$\frac{d f(\boldsymbol{x}, \boldsymbol{p}, t)}{d t}=0 \quad \text { where } \quad \frac{d}{d t}=\frac{\partial}{\partial t}+\dot{\boldsymbol{x}} \cdot \nabla_x+\dot{\boldsymbol{p}} \cdot \nabla_p$$

## 物理代写|宇宙学代写宇宙学代考|谐势中粒子的玻尔兹曼方程

$$E=\frac{p^2}{2 m}+\frac{1}{2} k x^2,$$

$$\dot{x}=\frac{p}{m} \quad \text { and } \quad \dot{p}=-k x .$$当推广到相对论情况时，这些熟悉的方程将被我们在2.1.2节中推导出的测地线方程所取代。此时，无碰撞玻尔兹曼方程为
$$\frac{\partial f}{\partial t}+\frac{p}{m} \frac{\partial f}{\partial x}-k x \frac{\partial f}{\partial p}=0 .$$

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