A Universal Soft-State Nonlocal Modification of Galactic Dynamics

A Universal Soft-State Nonlocal Modification of Galactic Dynamics Pham Xuan TungVietnam Abstract We propose a universal soft-state nonlocal model for galactic dynamics, motivated by hysteresis and memory effects in open gravitational systems. The model introduces a radially propagated state variable that modulates the effective MOND interpolation through a universal relaxation length expressed in units of the disk scale length. Using the SPARC rotation curve database (163 galaxies, 3271 data points with quality flag Q ≤ 2), we perform full Gaussian likelihood inference and compare directly against the standard MOND “simple” formulation. The soft-state model preserves both the Radial Acceleration Relation (RAR) and the Baryonic Tully–Fisher Relation (BTFR), while decisively outperforming MOND under AIC and BIC despite higher dimensionality. Profile likelihood analysis yields a sharply localized universal scale λ ≈ 0.40 R_d. Bootstrap and leave-one-galaxy-out diagnostics confirm parameter stability. Fully profiled confidence contours reveal mild covariance between the acceleration scale and the soft-state coupling but no multimodal structure. These results suggest that MOND phenomenology may emerge from hysteretic non-equilibrium dynamics with finite spatial memory, rather than from fundamental modification of gravity or the presence of dark matter at galactic scales. 1. Introduction Disk galaxies exhibit remarkably tight empirical relations between baryonic matter and observed kinematics, most notably the Radial Acceleration Relation (RAR) and the Baryonic Tully–Fisher Relation (BTFR). These relations pose a long-standing challenge to ΛCDM and have motivated alternative frameworks such as Modified Newtonian Dynamics (MOND). While MOND reproduces galactic phenomenology with surprising accuracy, it assumes an instantaneous and purely local mapping between baryonic acceleration and observed dynamics. However, many physical systems driven out of equilibrium exhibit hysteresis and memory effects, in which the current state depends on prior history. This motivates the hypothesis that galactic disks behave as open dynamical systems with finite spatial memory. In this work, we explore whether MOND-like behavior can emerge from a non-local soft-state mechanism characterized by a universal relaxation length. 2. Model Let y(r) = g_bar(r) / a0, where g_bar is the baryonic acceleration and a0 is a global acceleration scale. We define a binary target state h(y) with thresholds y_lo and y_hi: h = 1 if y ≥ y_hi h = 0 if y ≤ y_lo otherwise the state remains unchanged. A soft-state variable s(r) is propagated radially according to s(r_i) = h(r_i) + [s(r_{i-1}) − h(r_i)] exp(−Δr / λ), where λ is a relaxation length expressed in units of the disk scale length R_d. The observed acceleration is then given by g_obs(r) = g_bar(r) · ν(y(r) · exp(γ s(r))), where ν is the MOND “simple” interpolation function and γ controls the strength of the soft-state modulation. The model contains five global parameters: a0 (acceleration scale) y_lo, y_hi (hysteresis thresholds) γ (soft-state coupling) λ (non-local relaxation length). 3. Data and Methods We analyze the SPARC database, selecting galaxies with quality flag Q ≤ 2 (163 galaxies, 3271 rotation-curve points). Model predictions are compared directly to observed rotation velocities using a full Gaussian likelihood: −2 ln L = Σ [ (V_obs − V_model)^2 / σ^2 + ln(2πσ^2) ]. Optimization is performed using Powell minimization with warm starts. Model comparison is carried out using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). Parameter robustness is assessed via: bootstrap resampling over galaxies leave-one-galaxy-out diagnostics profile likelihood for λ fully profiled two-parameter confidence contours using sparse grids with spline interpolation. 4. Results 4.1 Global Fit The best-fitting parameters are approximately: log10 a0 ≈ 3.535log10 y_lo ≈ −0.91log10 y_hi ≈ 1.14γ ≈ −1.21λ ≈ 0.40 R_d Relative to MOND simple, the soft-state model achieves ΔAIC ≈ −3.2 × 10^4ΔBIC ≈ −3.2 × 10^4, indicating decisive statistical preference despite additional parameters. 4.2 RAR and BTFR The model preserves both the RAR and BTFR, reproducing their slopes and normalizations while reducing intrinsic scatter relative to MOND. 4.3 Constraint on the Non-local Scale Profile likelihood analysis shows that λ is sharply localized near 0.40 R_d, with a 3σ width of order ±0.02 R_d. This demonstrates that the non-local scale is a genuine inferred physical quantity rather than a weakly constrained nuisance parameter. 4.4 Parameter Stability Bootstrap and leave-one-galaxy-out tests indicate: a0 and γ are stable at the ~5–10% level λ remains tightly clustered moderate degeneracy exists in the threshold parameters. Fully profiled confidence contours in (a0, γ) show mild covariance but no evidence for multimodal solutions. 5. Discussion The emergence of a universal relaxation length suggests that galactic disks retain dynamical memory over approximately 0.4 scale lengths. This provides a physical interpretation of MOND-like phenomenology as an emergent property of hysteretic, non-equilibrium dynamics, rather than a fundamental modification of gravity or the presence of dark matter at galactic scales. In this view, scaling relations such as the RAR arise from state-dependent feedback between baryonic structure and effective dynamics. 6. Limitations and Future Work The present analysis assumes axisymmetric disks and neglects explicit time dependence. Future work should address: barred and interacting systems temporal evolution of the soft state cosmological initial conditions coupling to hydrodynamic feedback. Releasing a fully reproducible inference pipeline will be an important next step. 7. Conclusion We have demonstrated that a universal soft-state nonlocal extension of MOND provides a statistically superior description of SPARC rotation curves while preserving the RAR and BTFR. The existence of a sharply constrained relaxation scale supports the hypothesis that galactic dynamics encode finite spatial memory. Gravitational hysteresis offers a viable physical framework for galactic scaling laws without invoking dark matter. References Milgrom, M. (1983)Sanders & McGaugh (2002)McGaugh, Lelli & Schombert (2016)Lelli, McGaugh & Schombert (2016)