Preamble

This study introduces a path subspace to extend the energy phase space into a digital phase space (Λ-space), enabling the digital statistics of equilibrium systems composed of elastic particles. The Λ-space comprises three distinct regions corresponding to different modes of particle motion: translation, rotation, and vibration. For a system of N particles, we define three sets of basis vectors:

Let $E_k = \langle K_k, L_k, H_k \rangle$, $E_l = \langle L_l, H_l, K_l \rangle$, and $E_h = \langle H_h, K_h, L_h \rangle$ represent the state vectors for the three motion paths, where $K$, $L$, and $H$ denote the translational, rotational, and vibrational components respectively. The cyclic symmetry among these regions is expressed through the permutation of indices ${k, l, h}$.

The magnitude of each state vector is given by:
$$|E_z| = \sqrt{(H_z)^2 + (K_z)^2 + (L_z)^2}$$

where $z \in {k, l, h}$. The digital phase space Λ is thus partitioned as $\Lambda = \Lambda_k \cup \Lambda_l \cup \Lambda_h$, with each subspace characterized by its respective path variables.

2.3 Derived Energy

The derived energy framework establishes relationships between the path variables and system energy. For each region $z \in {k, l, h}$, we define derived energy components $P_z$ that satisfy:

$$P_z = \sqrt{(P_z^1)^2 + (P_z^2)^2 + (P_z^3)^2}$$

The derived energy operator connects the microscopic path variables to macroscopic thermodynamic quantities through the degree of association $C_n$, which characterizes particle interactions and avoids the potential energy integration problems inherent in conventional Γ-space formulations.

Table 1: Digitization of Energy and Pressure

[TABLE:1] Digitization of energy and pressure. The table establishes the fundamental relationships between the digital variables and physical quantities:

Parameter Region k Region l Region h Scale basis $a_s$ $a_k = N/(2C) = g_k$ $a_l = N/(2C) = g_l$ $a_h = N/(2C) = g_h$ Association $b_s$ $b_k = 1/(2a_k) = C/N = f_k$ $b_l = 1/(2a_l) = C/N = f_l$ $b_h = 1/(2a_h) = C/N = f_h$ Path energy $\tilde{K}_k = N$ $\tilde{L}_l = N$ $\tilde{H}_h = N$ Coupled energy $\tilde{L}_k = a_k N$ $\tilde{H}_l = a_l N$ $\tilde{K}_h = a_h N$ Interaction energy $\tilde{H}_k = b_k N$ $\tilde{K}_l = b_l N$ $\tilde{L}_h = b_h N$ Total energy $\tilde{E}_k = (a_k + b_k)N$ $\tilde{E}_l = (a_l + b_l)N$ $\tilde{E}_h = (a_h + b_h)N$ Volume factor $\tilde{V}^k = C$ $\tilde{V}^l = C$ $\tilde{V}^h = C$ Pressure factor $\tilde{P}^k = 2(a_k)^2 + 1$ $\tilde{P}^l = 2(a_l)^2 + 1$ $\tilde{P}^h = 2(a_h)^2 + 1$

4 Energy Relations and Thermodynamic Framework

4.1 Derived Energy System

The derived energy system expresses macroscopic thermodynamic quantities as functions of the digital variables. Table 2 summarizes the key relationships:

[TABLE:2] Derived energies of system and midson

Energy Component Expression $\tilde{E} = (a + b)N$ $\tilde{J} = (b - 1)N$ $\tilde{Q} = (1 + a)N$ $\tilde{G} = (a - b)N$ $\tilde{U} = (1 + a - b)N$ $\tilde{Y} = (1 + 2a - b)N$ $\tilde{\epsilon} = 2a^2 + 1$ $\tilde{\phi} = 1 - 2a$ $\tilde{\theta} = 2a(1 + a)$ $\tilde{\gamma} = 2a^2 - 1$ $\tilde{\upsilon} = 2a + 2a^2 - 1$ $\tilde{\psi} = 2a + 4a^2 - 1$

4.2 Reference State Vector

The reference state vector $k^* = \langle k, l, h \rangle = \langle 1, 1, 1 \rangle$ establishes the equilibrium condition where all three motion paths contribute equally to the system energy. This symmetric reference frame simplifies the analysis of phase transitions and critical phenomena.

4.3 Differential Relations

The thermodynamic differential equations in Λ-space generalize conventional thermodynamics through the introduction of path-specific differentials. The fundamental relation:

$$d\lambda = d(\theta - \eta) = d(\sigma E_s - P_2 V_s) = (E_s d\sigma - P_2 dV_s) - (-\sigma dE_s + V_s dP_2)$$

leads to the Gibbs-Duhem-type equation:

$$S dE_s - V dP_2 + C d\gamma = 0$$

This equation governs the equilibrium conditions across all three regions of the digital phase space.

4.4 Total Differential Equations

Table 3 presents the complete set of energy relations and their total differential forms, extending thermodynamic descriptions across all matter states:

[TABLE:3] Energy relations and total differential equations

Energy Relation Differential Form $E_1$ $E_1 = S E_s - P_2 V - C\gamma$ $dE_1 = E_s dS - P_2 dV - C d\gamma$ $E_2$ $E_2 = G + E_3 = Q - E_1 = P_2 V$ $dE_2 = S dE_s + P_2 dV + C d\gamma$ $E_3$ $E_3 = S E_s + P_2 V - \gamma C$ $dE_3 = S dE_s + P_2 dV - \gamma dC$ $Q$ $Q = E_1 + E_2 = U + E_3 = S E_s$ $dQ = E_s dS + V dP_2 - C d\gamma$ $G$ $G = E_2 - E_3 = U - E_1 = C\gamma$ $dG = -S dE_s + V dP_2 + \gamma dC$ $U$ $U = Q - E_3$ $dU = E_s dS - P_2 dV + \gamma dC$ $Y$ $Y = Q + G$ $dY = E_s dS + V dP_2 + \gamma dC$

The equilibrium condition is satisfied by:
$$S dE_s - V dP_2 + C d\gamma = 0$$

4.5 Physical Interpretation

The digital variables acquire physical meaning through association with measurable quantities. At reference temperature $T_2$, the association parameter $C$ equals the number of particles $N$, establishing a direct correspondence between the digital representation and physical reality. The scale basis $a(T)$ and association parameter $b(T)$ become temperature-dependent functions:

$$a(T) = \frac{V_A(T_2)}{2V_A(T)}, \quad b(T) = \frac{V_A(T)}{V_A(T_2)}$$

where $V_A(T)$ represents the molar volume at temperature $T$. The pressure equation:

$$P^* = (2a^2 + 1) \cdot \frac{kT}{N_A V_A(T_2)}$$

demonstrates how the digital parameters encode thermodynamic state information.

Physical Implementation and Validation

The theoretical framework is validated through analysis of elastic particle systems. Experimental observations show that elastic particle systems exhibit distinct behaviors: at low densities ($N \approx 18$), they display gas-like properties; at high densities ($N \approx 36$), they demonstrate liquid-solid characteristics. The transition region ($30^\circ C \sim 4^\circ C$) reveals critical phenomena where the digital parameters $a(T)$ and $b(T)$ undergo rapid variation, indicating structural transformations in the matter organization.

The Λ-space formulation successfully decodes structural information using four independent parameters: one digital variable ($C$) and three scale bases ($a_k, a_l, a_h$). This approach provides a unified statistical description that bridges microscopic particle dynamics with macroscopic thermodynamics while avoiding the divergences associated with traditional potential energy integration.

Abstract

By introducing the path subspace, this study extends the energy phase space to a digital phase space (Λ-space), thus accomplishing the digital statistics of the elastic particle equilibrium systems. Unlike the Γ phase space (6N dimensions) of Gibbs statistical ensemble, the Λ-space (18N dimensions) is divided into three regions: liquid, solid, and gas, and their energy probability distributions have cyclic symmetry. The interaction between particles in Λ-space is characterized by the degree of association, thereby avoiding the problem of potential energy integration in Γ-space. This article presents the macroscopic and mesoscopic equilibrium conditions for elastic particle systems and derives the partition functions and state functions for three regions. In addition, the thermodynamic differential equations are generalized to the entire range of matter states by extending the definition of the entropy function. The results indicate that the structural information of matter can be decoded using four independent parameters: one digital variable and three scale bases.

Keywords: statistical physics, elastic particles, statistical phase space, partition function, thermodynamical functions, structure of matter

Corresponding author. E-mail: zcliang@njupt.edu.cn