Distribution Functions and Transport Equations

A scalar phase-space distribution is justified when excitations occupy identifiable, sufficiently narrow spectral branches and internal coherence can be neglected or separately resolved. It is a coarse-grained description of a quantum state, not an exact observable in every interacting or gauge theory.

Required background. Use Hamiltonian and symplectic mechanics for phase-space flow and thermal spectral representations for the relation between excitations and spectral support.

Helpful background. Wigner transformation and gradient expansion explains how phase-space variables arise from two-point functions.

Relativistic distribution and invariant moments

For a positive-energy quasiparticle of mass $m$, use the invariant-measure convention of de Groot, van Leeuwen, and van Weert 1980, ch. II:

$$dP=\frac{g\,d^3\mathbf p}{(2\pi)^3E_{\mathbf p}}, \qquad E_{\mathbf p}=\sqrt{\mathbf p^2+m^2},$$

where $g$ counts only the internal states not already carried by species or matrix indices. With this convention,

$$N^\mu(x)=\int dP\,p^\mu f(x,\mathbf p), \qquad T^{\mu\nu}(x)=\int dP\,p^\mu p^\nu f(x,\mathbf p).$$

$N^0=g\int d^3p\,f/(2\pi)^3$ and $T^{00}=g\int d^3p\,E_{\mathbf p}f/(2\pi)^3$, confirming the normalization. Scattering formulas often use $d\Pi=d^3p/[(2\pi)^32E]$; mixing $dP$ and $d\Pi$ without the corresponding factor of two is a common rate error.

For a mean-field force, the covariant Boltzmann equation is schematically

$$p^\mu\partial_\mu f +\mathcal F^\mu(x,p)\frac{\partial f}{\partial p^\mu} =C[f].$$

The force must be tangent to the mass shell: $p_\mu\mathcal F^\mu=0$ when $p^2=m^2$ is fixed. For an Abelian Lorentz force, $\mathcal F^\mu=qF^{\mu\nu}p_\nu$ satisfies this automatically. If the mass depends on spacetime, the shell and force terms must be derived together; differentiating at fixed $E$ can miss a term.

Streaming as Liouville flow

In canonical coordinates, collisionless evolution is

$$\partial_t f+\dot{\mathbf x}\cdot\nabla_{\mathbf x}f +\dot{\mathbf p}\cdot\nabla_{\mathbf p}f=0.$$

Hamiltonian flow has zero phase-space divergence, so $f$ is constant along characteristics. For a free relativistic particle,

$$f(t,\mathbf x,\mathbf p) =f_0\!\left(\mathbf x-\frac{\mathbf p}{E_{\mathbf p}}(t-t_0),\mathbf p\right).$$

Direct substitution verifies the collisionless equation. Integrating over momentum gives a continuity equation if surface terms at $|\mathbf p|\to\infty$ vanish.

Taking moments of the full equation yields

$$\partial_\mu N^\mu=\int dP\,C[f], \qquad \partial_\mu T^{\mu\nu} =\int dP\,p^\nu C[f]+\text{mean-field exchange}.$$

Collision invariants make the appropriate right-hand sides vanish only after summing all coupled species and reactions. A species number need not be conserved even when total charge and energy–momentum are.

The hierarchy places a distribution downstream of a Wigner correlator and a controlled shell projection. This page begins with the reduced object; the diagram records the quantum and coherence information that has already been assumed negligible or separately retained.

A scalar distribution is a reduced description, not an arbitrary rewriting of a two-point function. Its phase-space moments and Liouville streaming are meaningful only after shell, gradient, state, and gauge-link conventions are controlled. Collision, entropy, and closure boxes are subsequent modeling steps, and microscopic matching may use cuts only in a controlled weak-coupling quasiparticle construction. Leading internal-state coherence instead follows the dashed matrix-valued branch. The diagram is schematic and not to scale.

The sections Relativistic distribution and invariant moments, Streaming as Liouville flow, and Quantum and gauge qualifications give the text and equation equivalent of the distribution-level assumptions.

Quantum and gauge qualifications

Bosonic and fermionic statistics enter collision terms through $1+f$ and $1-f$; they do not turn $f$ into an exact occupation operator. For gauge-charged particles, a gauge-covariant Wigner transform needs Wilson lines, and the distribution can carry color or flavor indices. Collective modes already included in mean fields must not also be counted as independent particles.

Finite spectral width replaces a sharp shell by a function of $p^0$. Particle–antiparticle or flavor coherence requires off-diagonal densities. In either case a scalar $f(x,\mathbf p)$ loses information, and the error must be estimated against a matrix or two-time description.

Failure tests

• Verify the measure by reconstructing $N^0$ and $T^{00}$.
• Vary the particle basis when spectral peaks overlap; stable observables should not depend strongly on an arbitrary diagonalization.
• Check $p\cdot\mathcal F=0$ and gauge covariance.
• Compare widths, branch separations, gradients, and memory times with the scales resolved by $f$.
• Integrate the numerical collision term against every conserved charge and $p^\mu$.
• Ensure degeneracies and collective modes are counted exactly once.

Exercise

Use the free-streaming solution to show that the total particle number $\int d^3x\,d^3p\,f$ is constant when $f$ vanishes at spatial infinity.

Solution

The solution translates $\mathbf x$ by a momentum-dependent constant. For each $\mathbf p$, change variables to $\mathbf y=\mathbf x-\mathbf v(t-t_0)$; the Jacobian is one, so the spatial integral equals its initial value. Equivalently, integrate the Liouville equation and discard the boundary flux.

Continue

Derive the distribution from two-time dynamics on Kadanoff–Baym to kinetic theory and build its interactions on collision kernels.

References

• Boltzmann, L. (1872). “Further Studies on the Thermal Equilibrium of Gas Molecules.” Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften 66, 275–370. English translation.
• de Groot, S. R., van Leeuwen, W. A., and van Weert, C. G. (1980). Relativistic Kinetic Theory: Principles and Applications. Amsterdam: North-Holland. WorldCat record.