Evolution of Jeans instability in nonthermal magnetized complex plasmas

The gist The study presents a fluid simulation investigating nonlinear evolution of Jeans instability in a magnetized tricomponent complex (dusty) plasma to understand early-stage star formation and evolution processes in nonthermal astrophysical media.

Physical Model and Linear Analysis

The physical model considers a self-gravitating, collisionless, and magnetized three component dusty plasma consisting of extremely massive, negatively charged dust grains along with highenergy superthermal electrons and ions Under the action of collective electrostatic, selfgravitational, and compressional magnetic forces, the hydrodynamic transport of the dust species is governed by the 1D fluid continuity and momentum equations To model the dynamic compression and stretching of the magnetic field lines perpendicular to the collapse direction, we self-consistently solve the magnetic induction equation under the ideal magnetohydrodynamic (MHD) frozen-in law The wave propagation is restricted to onedimensional (1D) Cartesian axis (x-axis), while a uniform ambient magnetic field is applied perpendicular to the direction of propagation (z-axis)

The background species are modeled using the 1D isotropic κ-distribution which has the following form Integrating this distribution function in velocity space yields the self-consistent electron and ion densities The resulting dimensionless equations are detailed in Table I.

Linear Dispersion Relation

Employing a plane-wave perturbation technique, with all perturbed quantities varying as exp(i˜kx˜−iω˜t˜), we linearize and Fourier transform Eqs. (9)–(15) to obtain the following linear dispersion relation for the magnetized Jeans instability Equation (16) shows that the compressional magnetic term (vA/vthi)2k squared enters additively and is independent of the nonthermal spectral index κ, in contrast to the electrostatic screening term, which is explicitly κ-dependent through ˜λe f f For instability (ω˜2 < 0), Eq. (16) requires αG to exceed the magnetized critical threshold αGc(˜k) = ˜k 2/˜k squared + ˜λ−2e f f+ωpi/ωpd 2vA/vthi squared, which increases monotonically with vA and hence Bz

Simulation Setup

To simulate the nonlinear spatiotemporal evolution of the self-gravitating, magnetized dusty plasma, we employ a highorder, self-consistent one-dimensional fluid simulation framework previously used in Refs. [18, 32] The first-order spatial derivatives in the dust continuity, momentum, and induction equations are approximated using a fourth-order accurate (∼ ∆x 4, ∆x is the grid step) central finite-difference method Temporal advancement is performed with the explicit leapfrog scheme, which is second-order accurate in time (∼ ∆t squared, ∆t being the time step) To achieve spatial uniformity with our fourth-order fluid solvers, we discretize the second-order spatial derivatives in Eqs. (12) and (13) using the stencil mentioned in Eq. (19) The resulting discrete electrostatic and gravitational Poisson equations for the SOR update are formulated as Eqs. (21)–(22) To suppress these numerical short-wavelength discretization errors, we pass the updated fluid and field variables through a fourthorder compensated low-pass filter at the end of every time step

Results and Discussion

The spatiotemporal evolution shows that during the early transient phase (ωpit = 0−12), the system undergoes an electrostatic decompression where outward fluid motion drives mass away from the center, resulting in a decrease of the central density peak As the system continues to evolve (ωpit = 150), all four fields now display a broad, substantially disturbed region flanking a sharp central spike Crucially, the ideal magnetohydrodynamic fluxfreezing condition forces the perpendicular magnetic field lines to be dragged inward and compressed along with the collapsing dust fluid This coupling results in a dramatic localized amplification of the magnetic field, which peaks at approximately 1.5, nearly four times its initial ambient strength, exactly in phase with the dust density core

The growth rate γsim is extracted directly from the slopes of these curves A systematic attenuation of the exponential growth with increasing Bz is clearly observed, as the unmagnetized case (Bz = 0) exhibits the largest growth; increasing Bz up to 1.2 progressively reduces the slope of the runaway phase At strong magnetic fields (Bz ≈ 2), this magnetic pressure becomes the dominant restoring force, completely erasing the kinetic electrostatic differences and forcing all κ distributions to converge to an identically suppressed growth rate

Conclusions

The main findings are:

1. Application of an ambient magnetic field significantly raises the effective stability threshold relative to the unmagnetized case, consistent with the compressional nature of the magnetic restoring force Nonlinear simulations confirm this directly as the simulated growth rate γsim falls by more than an order of magnitude as Bz increases from 0 to 2 (e.g., 0.0542 to 0.0024ωpi at κ = 2) for every κ examined

2. An increase in nonthermality (smaller κ) leads to faster structural localization However, the κ-dependence of the growth rate, which is clear in the unmagnetized case, is progressively suppressed with increasing magnetic field strength In strongly magnetized environments, the growth rates are largely insensitive to the degree of nonthermality This suggests that under sufficiently magnetized conditions, compressive magnetic pressure mainly governs the collapse rate, outweighing the effects of nonthermal thermodynamic statistics

These results extend our earlier unmagnetized study by showing that a modest ambient field can substantially delay or suppress Jeans-instability-driven collapse in nonthermal dusty plasmas This carries direct implications for structure formation in magnetized, nonthermal environments such as diffuse interstellar clouds where both effects are simultaneously present The paper concludes that in sufficiently magnetized environments, the thermodynamic state of the background plasma is entirely subdominant to the ambient magnetic field in regulating the onset of structural collapse.

The work is supported by acknowledgments from Tezpur University and APNDRL Data generated and used in this investigation are available upon reasonable request from the corresponding author. The simulation parameters were fixed for all runs, including a fixed value of αG = 0.1 which remains well above threshold across the entire range of Bz explored here. The growth rate falls from γsim ≈ 0.046–0.054ωpi in the unmagnetized limit (Bz = 0) to γsim ≈ 0.0024ωpi at Bz = 2.0, representing a suppression of more than an order of magnitude. The growth rates converge across all the κ values as Bz increases, suggesting the systematic narrowing of the κ-dependent spread in γsim. The study develops a comprehensive and numerically robust framework to understand early-stage star formation and evolution processes in nonthermal astrophysical media.

1. Improve simulations of star formation in astrophysical media by incorporating a comprehensive framework that accounts for nonthermal electron and ion distributions via k-distribution laws (Eqs 6, 7, 8). This allows AI systems to model structural evolution in environments like dense molecular clouds where standard thermal assumptions fail.

2. Enhance the capability of predictive models for gravitational collapse by implementing the derived linear dispersion relation (Eq. 16) into an AI system's stability analysis module. This enables the AI to predict critical magnetic field strengths and nonthermality levels that determine whether a structure will undergo Jeans instability or be stabilized, based on Eq. 17.

3. Develop a robust numerical solver for nonlinear fluid dynamics by integrating the fourth-order finite difference method (Eqs 18, 19) with a second-order leap-frog time integration scheme (Eq. 20). This allows the AI to perform self-consistent, long-term simulations tracking how magnetic pressure and electrostatic screening interact during the nonlinear runaway phase.

4. Implement advanced spatial filtering techniques, such as the fourth-order compensated low-pass filter (Eq. 23), within the simulation engine of an AI system. This suppresses numerical short-wavelength discretization errors while preserving physically meaningful long-wavelength Jeans waves, leading to more accurate long-term structural evolution predictions.

5. Create a parameter sensitivity analysis tool that maps the growth rate γsim against magnetic field strength Bz and spectral index κ (as shown in Fig 3). This tool allows the AI to quantitatively determine when the magnetic pressure becomes the dominant restoring force, predicting whether thermal or nonthermal plasma statistics are relevant for collapse regulation.