Phase space volume preserving dynamics for non-Hamiltonian systems

TL;DR

The paper tackles the problem of phase-space volume nonconservation in non-Hamiltonian dynamics by introducing a classical density-matrix framework in which the stability matrix ${\bm{A}}$ is split into a symmetric part ${\bm{A}}_+$ and an antisymmetric part ${\bm{A}}_-$. Norm-preserving tangent-space evolutions ${\widetilde{{\bm{M}}}}$ and ${\bm{M}}_-$ separate stretching from rotation, with ${\bm{M}}_-$ acting unitarily to prevent tangent-vector collapse; a volume-preserving operator ${\widetilde{{\bm{M}}}'}$ yields a time-invariant phase-space volume for mixed states. The generalized Liouville equation and invariant measures arise from these constructions, enabling computation of instantaneous Lyapunov exponents via multiple basis choices and a classical Bloch-sphere picture of tangent-space dynamics. The framework is demonstrated on the linear and damped harmonic oscillators, Hénon-Heiles, and Lorenz-Fetter, showing computational advantages and scalability to higher dimensions for analyzing chaotic, dissipative, and driven systems.

Abstract

Infinitesimal volumes stretch and contract as they coevolve with classical phase space trajectories according to a linearized dynamics. Unless these tangent space dynamics are modified, the underlying chaotic dynamics will cause the volume to vanish as tangent vectors collapse on the most expanding direction. Here, we propose an alternative linearized dynamics and rectify the generalized Liouville equation to preserve phase space volume, even for non-Hamiltonian systems. Within a classical density matrix theory, we define the time evolution operator from the anti-symmetric part of the stability matrix so that phase space volume is time-invariant. The operator generates orthogonal transformations without distorting volume elements, providing an invariant measure for dissipative dynamics and a evolution equation for the density matrix akin to the quantum mechanical Liouville-von Neumann equation. The compressibility of volume elements is determined by a non-orthogonal operator made from the symmetric part of the stability matrix. We analyze complete sets of basis vectors for the tangent space dynamics of chaotic systems, which may be dissipative, transient or driven, without re-orthogonalization of tangent vectors. The linear harmonic oscillator, the Lorenz-Fetter model, and the Hénon-Heiles system demonstrate the computation of the instantaneous Lyapunov exponent spectrum and the local Gibbs entropy flow rate using these bases and show that it is numerically convenient.

Figures (7)

• Figure 1: Time evolution of the initial pure states $\ket{\delta {\bm{u}}_1(t_0)}$ and $\ket{\delta {\bm{u}}_2(t_0)}$ at time $t_0$ to the states $\ket{\delta {\bm{u}}_1(t)}$ and $\ket{\delta {\bm{u}}_2(t)}$ at time $t$, under two norm-preserving dynamics governed by the evolution operators $\widetilde{{\bm{M}}}$ and ${\bm{M}}_-$ along a given classical trajectory. The operator ${\bm{M}}_-$ is orthogonal and therefore preserves both the norm and the angle $\theta$ between the two states over time. While $\widetilde{{\bm{M}}}$ preserves the norm, it does not preserve the angle.
• Figure 2: Time evolution of a set of basis states from an initial time $t_0$ to a time $t$ under two norm-preserving dynamics, $\widetilde{{\bm{M}}}$ and ${\bm{M}}_-$ along a given classical trajectory. However, unlike $\widetilde{{\bm{M}}}$, the operator ${\bm{M}}_-$ preserves the angles between the basis states. Their inner products are time invariant: $\bra{\delta {\bm{u}}_i(t)}\ket{\delta {\bm{u}}_j(t)} = \bra{\delta {\bm{u}}_i(t_0)}\ket{\delta {\bm{u}}_j(t_0)}$ under unitary evolution ${\bm{M}}_-$.
• Figure 3: Classical analogue of the Bloch sphere for ${\bm{M}}_-$ dynamics. --- Time evolution of an initial volume spanned by a set of complete basis vectors $\ket{\delta \bm{\phi}_j}$ along the Lorenz attractor. These basis states ${\bm{\varrho}}_j=\dyad{\delta \bm{\phi}_j}{\delta \bm{\phi}_j}$ define a unit 2-sphere whose volume remains invariant with time evolution under the ${\bm{M}}_-$ dynamics. This volume is defined at each phase space point. We compute these perturbation vectors along the trajectory by evolving a set of initial basis vectors at time $t_0$ using the ${\bm{M}}_-$ operator and show them at three phase space points. At any point, these perturbation vectors determine the local volume variation rate $\Lambda = \sum_{j=1}^N\Tr({\bm{A}}_+{\bm{\varrho}}_j)$. The three large spheres with basis vectors are zoomed in versions of the nearest small spheres on the trajectory.
• Figure 4: (a) Linear harmonic oscillator ($\omega=0.5$): $\Tr{\bm{\xi}}$ (black) and $\Tr{\bm{\varrho}}$ (gray) as a function of time, (b) Time evolution of instantaneous Lyapunov exponent (ILE) for an arbitrary pure state, and (c) ILEs in the conjugate tangent space directions.
• Figure 5: Damped harmonic oscillator ($\omega=0.5$, $\gamma=0.05$): (a) Time evolution of $\Tr{\bm{\xi}}$ (black) and $\Tr{\bm{\varrho}}$ (gray) and (b) instantaneous Lyapunov exponent (ILE) for an arbitrary pure state, and (c) ILEs in the conjugate tangent space directions.