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  1. Boltzmann Constant

    Linked via "phase space"

    Here, $\Omega$ (Omega) represents the thermodynamic probability, or the number of distinct, equally probable microscopic configurations (microstates) that correspond to a given macroscopic state (macrostate) of the system [4]. The constant $k_B$ acts as the dimensional scale factor, translating the dimensionless count of microstates ($\ln \Omega$) into the physical units of entropy (Joules per Kelvin). A higher value of $\Omega$ reflects a deeper uncertainty about the system's precise internal configuration, which the constant translates into…

  2. Classical Dynamics

    Linked via "phase space"

    Hamiltonian Mechanics and Phase Space
    Hamiltonian mechanics represents a further abstraction of the Lagrangian formalism, transitioning the focus from configuration space to phase space. The Hamiltonian, $H$, typically corresponds to the total energy of the system ($H = T + V$), provided the constraints are time-independent ([scleronomic](/entries/…

  3. Classical Dynamics

    Linked via "phase space"

    $$H(\mathbf{q}, \mathbf{p}, t) = \sumi pi \dot{q}_i - L$$
    where $pi$ are the generalized momenta, $pi = \frac{\partial L}{\partial \dot{q}_i}$. The evolution of the system in phase space is governed by Hamilton's canonical equations:
    $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$

  4. Classical Dynamics

    Linked via "phase space"

    $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$
    A crucial feature of Hamiltonian dynamics is the preservation of phase space volume under time evolution, as mandated by Liouville's Theorem. This theorem implies that the density of representative points in phase space remains constant along trajectories. This conservation is mathematically rigorous but practically complicated by the phenomen…

  5. Differential Equations

    Linked via "phase space"

    When considering DEs that model dynamic systems, stability analysis is paramount. The long-term behavior of solutions, rather than their transient phase, often reveals the essential nature of the underlying physical process.
    For systems modeled by first-order ODEs, the system trajectories in phase space (the state space) converge toward specific geometric structures known as attractors.
    | Attractor Type | Descripti…