---
title: 'Yukawa Transport Logarithm: Screened Interactions'
url: https://www.emergentmind.com/topics/yukawa-transport-logarithm
type: topic
---

# Yukawa Transport Logarithm: Screened Interactions

Searching arXiv for the cited paper and closely related work on Yukawa/generalized transport logarithms.
The **Yukawa transport logarithm** is the screened analogue of the usual gravitational or Coulomb logarithm that controls kinetic relaxation in systems with finite-range Yukawa interactions. In the Yukawa-screened Schrödinger–Poisson setting studied in "Yukawa-Screened Bose-Star Condensation" [2605.23206], it quantifies how a finite interaction range suppresses the infrared contribution of many small-angle scatterings that would otherwise generate the familiar logarithmic enhancement of relaxation in the unscreened Schrödinger–Poisson case. In that formulation, the ordinary factor \(\ln(mvR)\) is replaced by a finite transport integral \(\Lambda_Y\), and the condensation time of a bosonic gas is correspondingly modified [2605.23206]. Related plasma-kinetic literature uses closely analogous constructions—either as a generalized Coulomb logarithm for Debye-screened Yukawa scattering [1109.4826], as an effective Coulomb logarithm for one-component plasmas and weakly screened Yukawa matter [1304.7134], or as a generalized momentum-transfer coefficient built from screened cross sections in ion friction models [1905.08715]. These usages are related but not identical.

## 1. Definition in the Yukawa-screened Schrödinger–Poisson system

In the Yukawa-screened Schrödinger–Poisson system, the field equations are written as
\[
i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,
\]
\[
(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),
\]
with Yukawa two-body potential
\[
V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},
\]
and Fourier kernel
\[
\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.
\]
Within this framework, the Yukawa transport logarithm is introduced as the screened replacement for the unscreened gravitational Coulomb logarithm appearing in the condensation time [2605.23206].

The screened condensation-time formula is
\[
\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},
\]
while the unscreened result is
\[
\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.
\]
The paper makes the replacement explicit through
\[
\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},
\]
so the role of \(\Lambda_Y\) is exactly the role played by \(\ln(mvR)\), but modified by finite interaction range [2605.23206].

The central definition is
\[
\Lambda_Y = \int_{q_{\rm min}}^{q_{\rm max}} \frac{q^3\,dq}{(q^2+\mu_Y^2)^2},
\]
with
\[
q_{\rm min}=R^{-1},\qquad q_{\rm max}=mv.
\]
Evaluating the integral gives
\[
\Lambda_Y = \frac{1}{2} \left[ \ln \frac{m^2v^2+\mu_Y^2} {R^{-2}+\mu_Y^2} + \frac{\mu_Y^2}{m^2v^2+\mu_Y^2} - \frac{\mu_Y^2}{R^{-2}+\mu_Y^2} \right].
\]
The paper emphasizes that this quantity is not merely a logarithm of a cutoff ratio; it is a finite transport integral that reduces to the Coulomb logarithm only when \(\mu_Y\to 0\) [2605.23206].

## 2. Derivation from screened small-angle scattering

The derivation begins from small-angle Yukawa scattering. Using
\[
q=2mv\sin\frac{\theta}{2}\simeq mv\theta,
\]
the screening mass defines the angular scale
\[
\theta_Y=\frac{\mu_Y}{mv}.
\]
The differential cross section is written as
\[
\frac{d\sigma_Y}{d\Omega} = \frac{4G^2m^2}{v^4} \frac{1}{\left(\theta^2+\theta_Y^2\right)^2},
\qquad
\theta_Y\equiv\frac{\mu_Y}{mv},
\]
which reduces to the Rutherford form in the unscreened limit [2605.23206].

The relevant quantity for relaxation is not the total cross section but the transport cross section,
\[
\sigma_{\rm tr,Y} = \int d\Omega\,(1-\cos\theta) \frac{d\sigma_Y}{d\Omega}.
\]
Using the small-angle approximations
\[
1-\cos\theta\simeq\frac{\theta^2}{2},
\qquad
d\Omega\simeq2\pi\theta\,d\theta,
\]
the integral becomes
\[
\sigma_{\rm tr,Y}
=
\frac{4\pi G^2m^2}{v^4}
\int_{\theta_{\rm min}}^{\theta_{\rm max}}
\frac{\theta^3\,d\theta}{(\theta^2+\theta_Y^2)^2}.
\]
Changing variables to momentum transfer gives
\[
\frac{\theta^3\,d\theta}{(\theta^2+\theta_Y^2)^2}
=
\frac{q^3\,dq}{(q^2+\mu_Y^2)^2},
\]
and the transport cross section is written in the normalization used in the condensation literature as
\[
\sigma_{\rm tr,Y} = \frac{8\pi G^2m^2}{v^4}\Lambda_Y.
\]
This identifies \(\Lambda_Y\) as the screened version of the unscreened transport integral \(\int dq/q\) that yields the ordinary Coulomb logarithm [2605.23206].

The infrared and ultraviolet cutoffs are introduced as
\[
q_{\rm min}=R^{-1},
\qquad
q_{\rm max}=mv.
\]
Their interpretation is standard: \(q_{\rm min}=R^{-1}\) is a finite-size infrared cutoff, while \(q_{\rm max}=mv\) is the ultraviolet cutoff set by the typical particle momentum transfer in the gas. In the unscreened case the integrand behaves like \(dq/q\), producing \(\ln(mvR)\). With screening, the denominator \(q^2+\mu_Y^2\) regulates small-\(q\) scattering and makes the transport factor finite [2605.23206].

## 3. Physical meaning and limiting regimes

The physical content of the Yukawa transport logarithm is that finite interaction range suppresses long-wavelength, small-angle scattering. For a \(1/r\) force, the Fourier kernel scales as \(1/q^2\), so the transport weighting produces an integral of the form
\[
\int \frac{dq}{q},
\]
which accumulates equally from each logarithmic interval and yields the Coulomb logarithm. In that case, the enhancement comes from the infrared sector, namely many weak long-range deflections [2605.23206].

For the Yukawa potential, the kernel instead has the form
\[
\widetilde V_Y(q)\propto \frac{1}{q^2+\mu_Y^2}.
\]
For \(q\ll \mu_Y\), the kernel saturates rather than diverges. In real space, the corresponding statement is that beyond the screening length
\[
\ell_Y \sim \mu_Y^{-1},
\]
the force is exponentially suppressed. This means that scatterings with impact parameters larger than \(\ell_Y\) are ineffective, and the infrared contribution is cut off by the interaction range itself rather than only by the system size [2605.23206].

The competition among the screening scale \(\mu_Y^{-1}\), the system size \(R\), and the typical momentum scale \(mv\) determines the size of \(\Lambda_Y\). The paper states the following trends. When \(\mu_Y \ll R^{-1}\), screening is irrelevant on system scales and Newtonian behavior is recovered. When \(R^{-1}\ll \mu_Y \ll mv\), the lower part of the logarithmic interval is removed and relaxation is weakened. When \(\mu_Y \gtrsim mv\), even momentum transfers of order the particle momentum are screened, so the transport logarithm becomes small and condensation slows strongly [2605.23206].

The unscreened limit is
\[
\Lambda_Y \rightarrow \ln(mvR)
\qquad \text{as } \mu_Y\rightarrow0.
\]
This limit is central: it shows that the Yukawa transport logarithm is a finite-range deformation of the familiar Coulomb logarithm rather than an unrelated object [2605.23206].

## 4. Role in kinetic relaxation and Bose-star condensation

In the Yukawa-screened Bose-star problem, the condensation time is tied to the relaxation time of a highly occupied bosonic gas. The relaxation estimate is written as
\[
\tau = \frac{4b\sqrt{2}}{\sigma_{\rm tr} v n f},
\]
where \(b=O(1)\), \(n\) is the number density, \(v\) is the characteristic velocity, and \(f\) is the occupation number. For an isotropic distribution with momentum width \(mv\),
\[
f=\frac{6\pi^2 n}{(mv)^3}.
\]
Substituting the screened transport cross section into this estimate yields
\[
\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{m v^6}{G^2n^2\Lambda_Y}.
\]
Hence
\[
\tau_Y \propto \frac{1}{\Lambda_Y}.
\]
Smaller \(\Lambda_Y\) means slower kinetic relaxation and a longer delay before condensation [2605.23206].

The paper validates this screened kinetic scaling numerically. Simulations are performed for homogeneous, isotropic random-wave initial conditions in a periodic box, and the Yukawa screening parameter \(\widetilde\mu_Y\) is varied in the range \(0\le \widetilde\mu_Y\le 1\). Increasing \(\widetilde\mu_Y\) delays the rise of the maximum density and thus delays Bose-star formation. Measured condensation times are compared with the theoretical prediction from the screened formula depending on \(\Lambda_Y\), keeping the analytic scaling fixed and fitting only the overall normalization coefficient. The best-fit value reported is
\[
b_{\rm fit} \approx 0.54,
\]
which the paper notes is close to \(b\simeq0.6\) found for Bose-star condensation with gravity. The reported agreement between theory and numerics across different box sizes supports the use of \(\Lambda_Y\) as the transport factor controlling condensation time [2605.23206].

A common misconception is to treat the screened replacement as simply \(\ln[(mv)/\mu_Y]\). The paper explicitly distinguishes its result from that simplification: the screened replacement is the finite transport integral \(\Lambda_Y\), not merely a logarithm of two scales [2605.23206]. This suggests that precision applications should retain the full closed form rather than substitute a rough cutoff estimate.

## 5. Relation to generalized Coulomb logarithms in plasma transport

A broader kinetic-theory literature uses closely related objects for screened Coulomb or Yukawa plasmas, but with different observables and averaging procedures. In "A generalized Coulomb logarithm for strongly coupled plasmas" [1109.4826], the generalized transport logarithm is
\[
\Xi = \frac{1}{2}\int_0^\infty d\xi\, e^{-\xi^2}\,\xi^5\, \frac{\sigma_s(\xi,\Lambda)}{\sigma_o},
\]
where \(\sigma_s\) is the momentum-transfer cross section. This \(\Xi\) is a thermal average over exact Yukawa binary scattering and replaces \(\ln\Lambda\) in collision frequencies governing friction, temperature relaxation, and resistivity [1109.4826].

That generalized Coulomb logarithm differs conceptually from \(\Lambda_Y\) in the Bose-star problem. The plasma quantity \(\Xi\) is a Maxwellian average of a screened momentum-transfer cross section, whereas \(\Lambda_Y\) is the transport integral entering a small-angle condensation-time estimate in the Yukawa-screened Schrödinger–Poisson system. The shared idea is the same replacement
\[
\ln\Lambda \to \text{screened transport factor},
\]
but the detailed definitions are not interchangeable [1109.4826; 2605.23206].

A second related construction appears in "Effective Coulomb Logarithm for One Component Plasma" [1304.7134], which proposes
\[
\Lambda_{\rm eff} = \frac{1}{2}\ln\!\left[1+\left(\frac{\lambda_{\rm D}}{h}\right)^2\right]
\]
with
\[
\frac{h}{\lambda_{\rm D}} = \left[1+(3\Gamma)^{3/2}\right]^{1/3}-1.
\]
That paper does not derive a separate explicit Yukawa logarithm \(\Lambda_Y(\Gamma,\kappa)\). Instead, it argues that in weakly screened, strongly coupled Yukawa systems the reduced diffusion can be transferred from one-component-plasma scaling through the melting-line ratio \(\Gamma/\Gamma_{\rm M}\) [1304.7134]. A plausible implication is that the term “Yukawa transport logarithm” has been used in more than one technical sense: sometimes as an explicitly screened momentum-transfer integral, sometimes as an effective transport parameter inferred from related plasma scaling.

## 6. Alternative formulations, regimes of validity, and limitations

Several related works clarify where transport-logarithm language is useful and where it is not. In ion-ion friction at small values of the Coulomb logarithm, one implementation uses a generalized momentum-transfer coefficient
\[
\Xi(\Delta \overline{V })=\frac{3}{16 } \frac{1}{\Delta \overline{ V}^3} \frac{1}{2}
\int_0^{\infty} d \xi \, \xi^2 \frac{\sigma_{ss'}^{(1)}(\xi)}{\sigma_0} {\cal X},
\]
with the first momentum-transfer cross section \(\sigma_{ss'}^{(1)}\) taken from a screened Coulomb potential. In that usage, the relevant Yukawa transport quantity is again a generalized momentum-transfer coefficient rather than a simple cutoff logarithm [1905.08715]. This reinforces the broader point that screened transport factors are typically cross-section-based objects.

By contrast, "Auto-correlations of Microscopic Density Fluctuations for Yukawa Fluids in the Generalized Hydrodynamics Framework" [2209.05762] does not define or use a transport logarithm at all. Transport enters there through thermal diffusivity, longitudinal viscosity combination, relaxation time, sound attenuation, and sound speed in a generalized-hydrodynamic description of strongly coupled Yukawa fluids. Screening appears through rational factors such as
\[
\frac{\omega_p^2}{k^2+\lambda_D^{-2}},
\]
not through a logarithm [2209.05762]. This indicates that not every Yukawa transport theory is logarithmic; in strongly coupled regimes, collective hydrodynamic or viscoelastic descriptions may replace binary-collision logarithmics.

A further caveat comes from "Validation of Classical Transport Cross Section for Ion-Ion Interactions Under Repulsive Yukawa Potential" [2401.11891]. That work does not define a standalone Yukawa transport logarithm, but it shows that classical and quantum transport cross sections under a repulsive Yukawa potential agree only in an intermediate velocity window. It gives high-velocity asymptotic forms from which an effective screened transport logarithm could be inferred, but also shows that classical transport fails at both low and high velocities [2401.11891]. This suggests that any Yukawa transport logarithm built from classical cross sections is regime-dependent.

An additional limitation appears in heavy-quark transport beyond leading logarithm. There the logarithm arises from screened soft exchange integrated between \(m_D\) and \(T\), but the full leading-order dynamics are non-Gaussian and require the full momentum-transfer kernel rather than only drag and diffusion coefficients. The paper therefore shows that a single transport logarithm captures only the Gaussian core of the process, not the full equilibration dynamics [2604.21895]. A plausible implication is that Yukawa transport logarithms are most informative when transport is accurately summarized by momentum-transfer rates, and less complete when higher cumulants or non-Gaussian tails are dynamically essential.

## 7. Conceptual summary

In its most specific and explicit sense, the Yukawa transport logarithm is the finite transport integral
\[
\Lambda_Y = \int_{R^{-1}}^{mv}\frac{q^3\,dq}{(q^2+\mu_Y^2)^2}
\]
introduced in the Yukawa-screened Bose-star condensation problem, with closed form
\[
\Lambda_Y = \frac{1}{2} \left[ \ln \frac{m^2v^2+\mu_Y^2} {R^{-2}+\mu_Y^2} + \frac{\mu_Y^2}{m^2v^2+\mu_Y^2} - \frac{\mu_Y^2}{R^{-2}+\mu_Y^2} \right].
\]
It replaces the unscreened \(\ln(mvR)\), enters the transport cross section as
\[
\sigma_{\rm tr,Y} = \frac{8\pi G^2m^2}{v^4}\Lambda_Y,
\]
and lengthens the condensation time according to
\[
\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y}
\]
[2605.23206].

More generally, the phrase denotes a family of screened transport factors that replace the weak-coupling Coulomb logarithm in systems with Yukawa interactions. Across the literature, these factors may be defined as a momentum-transfer integral over \(q\), a thermal average of a Yukawa momentum-transfer cross section, or an effective Coulomb logarithm incorporating correlation-hole physics [2605.23206; 1109.4826; 1304.7134]. The common physical content is the same: finite-range screening suppresses the infrared accumulation of many weak deflections and thereby reduces transport relative to the unscreened long-range case.

The principal distinction from the ordinary Coulomb logarithm is therefore not merely algebraic but structural. The unscreened quantity is a logarithmic sensitivity to infrared and ultraviolet cutoffs; the Yukawa replacement is a screened transport factor whose finiteness arises from the interaction kernel itself. In that sense, the Yukawa transport logarithm is best understood as the finite-range transport factor replacing the ordinary Coulomb logarithm.

Source: https://www.emergentmind.com/topics/yukawa-transport-logarithm