Yukawa Transport Logarithm: Screened Interactions

• Yukawa transport logarithm is a finite transport integral that replaces the unscreened Coulomb logarithm in systems with finite-range Yukawa interactions.
• It quantifies how Yukawa screening suppresses long-range, small-angle scatterings by regulating the infrared divergence inherent in Coulomb interactions.
• This formulation impacts kinetic relaxation and Bose-star condensation by lengthening condensation times, with numerical simulations corroborating the theoretical predictions.

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" (Chen, 22 May 2026), 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 (Chen, 22 May 2026). Related plasma-kinetic literature uses closely analogous constructions—either as a generalized Coulomb logarithm for Debye-screened Yukawa scattering (Baalrud, 2011), as an effective Coulomb logarithm for one-component plasmas and weakly screened Yukawa matter (Khrapak, 2013), or as a generalized momentum-transfer coefficient built from screened cross sections in ion friction models (Sprenkle et al., 2019). 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 (Chen, 22 May 2026).

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 $\Lambda_Y$0, but modified by finite interaction range (Chen, 22 May 2026).

The central definition is

$\Lambda_Y$1

with

$\Lambda_Y$2

Evaluating the integral gives

$\Lambda_Y$3

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 $\Lambda_Y$4 (Chen, 22 May 2026).

2. Derivation from screened small-angle scattering

The derivation begins from small-angle Yukawa scattering. Using

$\Lambda_Y$5

the screening mass defines the angular scale

$\Lambda_Y$6

The differential cross section is written as

$\Lambda_Y$7

which reduces to the Rutherford form in the unscreened limit (Chen, 22 May 2026).

The relevant quantity for relaxation is not the total cross section but the transport cross section,

$\Lambda_Y$8

Using the small-angle approximations

$\Lambda_Y$9

the integral becomes

$$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$0

Changing variables to momentum transfer gives

$$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$1

and the transport cross section is written in the normalization used in the condensation literature as

$$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$2

This identifies $$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$3 as the screened version of the unscreened transport integral $$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$4 that yields the ordinary Coulomb logarithm (Chen, 22 May 2026).

The infrared and ultraviolet cutoffs are introduced as

$$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$5

Their interpretation is standard: $$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$6 is a finite-size infrared cutoff, while $$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$7 is the ultraviolet cutoff set by the typical particle momentum transfer in the gas. In the unscreened case the integrand behaves like $$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$8, producing $$i\frac{\partial \psi}{\partial t} = -\frac{1}{2m}\nabla^2\psi +m\Phi\psi,$$9. With screening, the denominator $$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$0 regulates small-$$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$1 scattering and makes the transport factor finite (Chen, 22 May 2026).

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 $$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$2 force, the Fourier kernel scales as $$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$3, so the transport weighting produces an integral of the form

$$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$4

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 (Chen, 22 May 2026).

For the Yukawa potential, the kernel instead has the form

$$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$5

For $$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$6, the kernel saturates rather than diverges. In real space, the corresponding statement is that beyond the screening length

$$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$7

the force is exponentially suppressed. This means that scatterings with impact parameters larger than $$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$8 are ineffective, and the infrared contribution is cut off by the interaction range itself rather than only by the system size (Chen, 22 May 2026).

The competition among the screening scale $$(\nabla^2-\mu_Y^2)\Phi = 4\pi Gm \left( |\psi|^2-n \right),$$9, the system size $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$0, and the typical momentum scale $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$1 determines the size of $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$2. The paper states the following trends. When $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$3, screening is irrelevant on system scales and Newtonian behavior is recovered. When $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$4, the lower part of the logarithmic interval is removed and relaxation is weakened. When $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$5, even momentum transfers of order the particle momentum are screened, so the transport logarithm becomes small and condensation slows strongly (Chen, 22 May 2026).

The unscreened limit is

$V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$6

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 (Chen, 22 May 2026).

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

$V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$7

where $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$8, $V_Y(r)=-\frac{Gm^2}{r}e^{-\mu_Y r},$9 is the number density, $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$0 is the characteristic velocity, and $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$1 is the occupation number. For an isotropic distribution with momentum width $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$2,

$$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$3

Substituting the screened transport cross section into this estimate yields

$$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$4

Hence

$$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$5

Smaller $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$6 means slower kinetic relaxation and a longer delay before condensation (Chen, 22 May 2026).

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 $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$7 is varied in the range $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$8. Increasing $$\widetilde V_Y(q) = -\frac{4\pi Gm^2}{q^2+\mu_Y^2}.$$9 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 $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$0, keeping the analytic scaling fixed and fitting only the overall normalization coefficient. The best-fit value reported is

$$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$1

which the paper notes is close to $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$2 found for Bose-star condensation with gravity. The reported agreement between theory and numerics across different box sizes supports the use of $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$3 as the transport factor controlling condensation time (Chen, 22 May 2026).

A common misconception is to treat the screened replacement as simply $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$4. The paper explicitly distinguishes its result from that simplification: the screened replacement is the finite transport integral $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$5, not merely a logarithm of two scales (Chen, 22 May 2026). 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" (Baalrud, 2011), the generalized transport logarithm is

$$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$6

where $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$7 is the momentum-transfer cross section. This $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$8 is a thermal average over exact Yukawa binary scattering and replaces $$\tau_Y = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\Lambda_Y},$$9 in collision frequencies governing friction, temperature relaxation, and resistivity (Baalrud, 2011).

That generalized Coulomb logarithm differs conceptually from $$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$0 in the Bose-star problem. The plasma quantity $$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$1 is a Maxwellian average of a screened momentum-transfer cross section, whereas $$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$2 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

$$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$3

but the detailed definitions are not interchangeable [(Baalrud, 2011); (Chen, 22 May 2026)].

A second related construction appears in "Effective Coulomb Logarithm for One Component Plasma" (Khrapak, 2013), which proposes

$$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$4

with

$$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$5

That paper does not derive a separate explicit Yukawa logarithm $$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$6. 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 $$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$7 (Khrapak, 2013). 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

$$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$8

with the first momentum-transfer cross section $$\tau_{\rm gr} = \frac{b\sqrt{2}}{12\pi^3} \frac{mv^6}{G^2n^2\ln(mvR)}.$$9 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 (Sprenkle et al., 2019). 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" (Dhaka et al., 2022) 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

$\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$0

not through a logarithm (Dhaka et al., 2022). 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" (Hu et al., 2024). 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 (Hu et al., 2024). 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 $\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$1 and $\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$2, 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 (Plessis et al., 23 Apr 2026). 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

$\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$3

introduced in the Yukawa-screened Bose-star condensation problem, with closed form

$\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$4

It replaces the unscreened $\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$5, enters the transport cross section as

$\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$6

and lengthens the condensation time according to

$\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$7

(Chen, 22 May 2026).

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 $\tau_Y = \tau_{\rm gr} \frac{\ln(mvR)}{\Lambda_Y},$8, a thermal average of a Yukawa momentum-transfer cross section, or an effective Coulomb logarithm incorporating correlation-hole physics [(Chen, 22 May 2026); (Baalrud, 2011); (Khrapak, 2013)]. 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.