Talks

Nonlinear dynamics aims to elucidate the basic mechanisms necessary to reflect the temporal behavior of a natural system. The data analysis and modeling techniques proposed by artificial intelligence (deep networks, computational reservoirs, recurrent networks, as examples), on the other hand, ostensibly resign the mechanistic vision for a data-oriented modeling paradigm.  In these lectures, these apparently antagonistic approaches will be analyzed in parallel, using as examples my work on the physics of birdsong production and vocal learning.

Small systems consisting of a few particles are increasingly technologically relevant. In such systems, an intense debate in microcanonical statistical mechanics has been about the correctness of Boltzmann’s surface entropy versus Gibbs’ volume entropy. While the former considers states within a fixed energy window centered around the energy of the system, the latter considers all states with energy lesser than or equal to the energy of the system. Both entropies have shortcomings─while Boltzmann entropy predicts unphysical negative/infinite absolute temperatures for small systems with an unbounded energy spectrum, Gibbs entropy entirely disallows negative absolute temperatures, in disagreement with experiments. We consider a relative energy window, motivated by the Heisenberg energy-time uncertainty principle and an eigenstate thermalization time inversely proportional to the system energy. The resulting entropy ensures positive, finite temperatures for systems without a maximum limit on their energy and allows negative absolute temperatures in bounded energy spectrum systems, e.g., with population inversion. It also closely matches canonical ensemble predictions for prototypical systems, for instance, correctly describing the zero-point energy of an isolated quantum harmonic oscillator. Overall, we enable accurate thermodynamic models for isolated systems with few degrees of freedom.

We study the dynamics of an athermal inertial active particle moving in a shear-thinning medium in $d=1$. The viscosity of the medium is modeled using a Coulomb-tanh function, while the activity is represented by an asymmetric dichotomous noise with strengths $-\Delta$ and $\mu\Delta$, transitioning between these states at a rate $\lambda$. Starting from the Fokker-Planck~(FP) equation for the time-dependent probability distributions $P(v,-\Delta,t)$ and $P(v,\mu\Delta,t)$ of the particle's velocity $v$ at time $t$, moving under the influence of active forces $-\Delta$ and $\mu\Delta$ respectively, we analytically derive the steady-state velocity distribution function $P_s(v)$, explicitly dependent on $\mu$. Also, we obtain a quadrature expression for the effective diffusion coefficient $D_e$ for the symmetric active force case~($\mu=1$). For a given $\Delta$ and $\mu$, we show that $P_s(v)$ exhibits multiple transitions as $\lambda$ is varied. Subsequently, we numerically compute $P_s(v)$, the mean-squared velocity $\langle v^2\rangle(t)$, and the diffusion coefficient $D_e$ by solving the particle's equation of motion, all of which show excellent agreement with the analytical results in the steady-state. Finally, we examine the universal nature of the transitions in $P_s(v)$ by considering an alternative functional form of medium's viscosity that also capture the shear-thinning behavior.

Hard-core lattice-gas models serve as minimal yet powerful models to explore entropy-driven phase transitions. In these models, particles are restricted from occupying neighboring sites up to a specified kth next-nearest neighbor, effectively bridging the behavior from simple nearest-neighbor exclusion to the continuum hard-sphere gas. While most prior studies have examined cases up to k = 3, this talk presents a detailed investigation of the lattice-gas model on a triangular lattice up to k = 7, using a rejection free flat histogram algorithm enhanced with cluster moves. Our findings reveal that for k = 3 to k = 7, the system exhibits a single, discontinuous phase transition from a low-density disordered fluid to a high-density sublattice-ordered phase. This conclusion is supported by the analysis of partition function zeros and the nonconvexity of entropy.

The basal ganglia are known to influence action selection and modulation of movement vigor, but whether and how they contribute to specifying the kinematics of learned motor skills is not understood. Here, we probe this question by recording and manipulating basal ganglia activity in rats trained to generate complex task-specific movement patterns with rich kinematic structure. We find that the sensorimotor arm of the basal ganglia circuit is crucial for generating the detailed movement patterns underlying the acquired motor skills. Furthermore, the neural representations in the striatum, and the control function they subserve, do not depend on inputs from the motor cortex. Taken together, these results extend our understanding of the basal ganglia by showing that they can specify and control the fine-grained details of learned motor skills through their interactions with lower-level motor circuits.

We uncover activity-driven crossover from phase separation to a new turbulent state in a two- dimensional system of counter-rotating spinners. We study the statistical properties of this active- rotor turbulence using the active-rotor Cahn-Hilliard-Navier-Stokes model, and show that the vor- ticity ω ∝ ϕ, the scalar field that distinguishes regions with different rotating states. We explain this intriguing proportionality theoretically, and we characterize power-law energy and concentra- tion spectra, intermittency, and flow-topology statistics. We suggest biological implications of such turbulence. This work has been done with Biswajit Maji and Nadia Bihari Padhan. https://arxiv.org/abs/2503.03843