The Boltzmann transport equation for general systems in the semiclassical limit gives, for a Fermi liquid,

${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {\partial E}{\partial {\vec {p}}}}\cdot {\frac {\partial f}{\partial {\vec {x}}}}-{\frac {\partial E}{\partial {\vec {x}}}}\cdot {\frac {\partial f}{\partial {\vec {p}}}}={\text{St}}[f]}$ ,

where ${\displaystyle f({\vec {p}},{\vec {x}},t)=f_{0}({\vec {p}})+\delta f({\vec {p}},{\vec {x}},t)}$  is the density of quasiparticles (here we ignore spin) with momentum ${\displaystyle {\vec {p}}}$  and position ${\displaystyle {\vec {x}}}$  at time ${\displaystyle t}$ , and ${\displaystyle E({\vec {p}},{\vec {x}},t)=E_{0}({\vec {p}})+\delta E({\vec {p}},{\vec {x}},t)}$  is the energy of a quasiparticle of momentum ${\displaystyle {\vec {p}}}$  (${\displaystyle f_{0}}$  and ${\displaystyle E_{0}}$  denote equilibrium distribution and energy in the equilibrium distribution). The semiclassical limit assumes that ${\displaystyle f}$  fluctuates with angular frequency ${\displaystyle \omega }$  and wavelength ${\displaystyle \lambda =2\pi /k}$ , which are much lower than ${\displaystyle E_{\rm {F}}/\hbar }$  and much longer than ${\displaystyle \hbar /p_{\rm {F}}}$  respectively, where ${\displaystyle E_{\rm {F}}}$  and ${\displaystyle p_{\rm {F}}}$  are the Fermi energy and momentum respectively, around which ${\displaystyle f}$  is nontrivial. To first order in fluctuation from equilibrium, the equation becomes

${\displaystyle {\frac {\partial \delta f}{\partial t}}+{\frac {\partial E_{0}}{\partial {\vec {p}}}}\cdot {\frac {\partial \delta f}{\partial {\vec {x}}}}-{\frac {\partial \delta E}{\partial {\vec {x}}}}\cdot {\frac {\partial f_{0}}{\partial {\vec {p}}}}={\text{St}}[f]}$ .

When the quasiparticle's mean free path ${\displaystyle \ell \ll \lambda }$  (equivalently, relaxation time ${\displaystyle \tau \ll 1/\omega }$ ), ordinary sound waves ("first sound") propagate with little absorption. But at low temperatures ${\displaystyle T}$  (where ${\displaystyle \tau }$  and ${\displaystyle \ell }$  scale as ${\displaystyle T^{-2}}$  ), the mean free path exceeds ${\displaystyle \lambda }$ , and as a result the collision functional ${\displaystyle {\text{St}}[f]\approx 0}$ . Zero sound occurs in this collisionless limit.

In the Fermi liquid theory, the energy of a quasiparticle of momentum ${\displaystyle {\vec {p}}}$  is

${\displaystyle E_{\rm {F}}+v_{\rm {F}}(|{\vec {p}}|-p_{\rm {F}})+\int {\frac {d^{3}{\vec {p}}'}{4\pi p_{\rm {F}}m^{*}}}F(p,p')\delta f(p')}$ ,

where ${\displaystyle F}$  is the appropriately normalized Landau parameter, and

${\displaystyle f_{0}({\vec {p}})=\Theta (p_{\rm {F}}-|{\vec {p}}|)}$ .

The approximated transport equation then has plane wave solutions

${\displaystyle \delta f({\vec {p}},{\vec {x}},t)=\delta (E({\vec {p}})-E_{\rm {F}})e^{i({\vec {k}}\cdot {\vec {r}}-\omega t)}\nu ({\hat {p}})}$ ,

with ${\displaystyle \nu ({\hat {p}})}$ ^[5] given by

${\displaystyle (\omega -v_{\rm {F}}{\hat {p}}\cdot {\hat {k}})\nu ({\hat {p}})=v_{\rm {F}}{\hat {p}}\cdot {\hat {k}}\int d^{2}{\frac {{\hat {p}}'}{4\pi }}F({\hat {p}},{\hat {p}}')\nu ({\hat {p}}')}$ .

This functional operator equation gives the dispersion relation for the zero sound waves with frequency ${\displaystyle \omega }$  and wave vector ${\displaystyle {\vec {k}}}$  . The transport equation is valid in the regime where ${\displaystyle \hbar \omega \ll E_{\rm {F}}}$  and ${\displaystyle \hbar |{\vec {k}}|\ll p_{\rm {F}}}$ .

In many systems, ${\displaystyle F({\hat {p}},{\hat {p}}')}$  only slowly depends on the angle between ${\displaystyle {\hat {p}}}$  and ${\displaystyle {\hat {p}}'}$ . If ${\displaystyle F}$  is an angle-independent constant ${\displaystyle F_{0}}$  with ${\displaystyle F_{0}>0}$  (note that this constraint is stricter than the Pomeranchuk instability) then the wave has the form ${\displaystyle \nu ({\hat {p}})\propto ({\omega }/({v_{\rm {F}}{\hat {p}}\cdot {\vec {k}}})-1)^{-1}}$  and dispersion relation ${\displaystyle {\frac {s}{2}}\log {\frac {s+1}{s-1}}-1=1/F_{0}}$  where ${\displaystyle s=\omega /{kv_{\rm {F}}}}$  is the ratio of zero sound phase velocity to Fermi velocity. If the first two Legendre components of the Landau parameter are significant, ${\displaystyle F({\hat {p}},{\hat {p}}')=F_{0}+F_{1}{\hat {p}}\cdot {\hat {p}}'}$  and ${\displaystyle F_{1}>6}$ , the system also admits an asymmetric zero sound wave solution ${\displaystyle \nu ({\hat {p}})\propto {\sin(2\theta )}/({s-\cos {\theta }})e^{i\phi }}$  (where ${\displaystyle \phi }$  and ${\displaystyle \theta }$  are the azimuthal and polar angle of ${\displaystyle {\hat {p}}}$  about the propagation direction ${\displaystyle {\hat {k}}}$ ) and dispersion relation

${\displaystyle \int _{0}^{\pi }{\frac {\sin ^{3}\theta \cos \theta }{s-\cos \theta }}d\theta ={\frac {4}{F_{1}}}}$ .

• Second sound
• Third sound

• Piers Coleman (2016). Introduction to Many-Body Physics (1st ed.). Cambridge University Press. ISBN 9780521864886.