Generalized Temperature Definitions

A variety of thermodynamic temperature definitions complementary to the classic kinetic definition have been derived recently (1,3,2). The most general form, proved in Ref. (3) and (4), is:

$$k_B T = \frac{\left< \nabla \ensuremath{{\mathcal H}({\mathbf \Ga... ...f \Gamma}) \right>}{\left< \nabla \cdot {\mathbf B}({\mathbf \Gamma}) \right>},$$ (1)

where ${\mathbf \Gamma}= \{\Gamma_{1}, \cdots , \Gamma_{6N}\} = \{q_1, \cdots , q_{3N}, p_1, \cdots , p_{3N}\}$ is the set of $3N$ generalized coordinates $q_i$ and $3N$ conjugate momenta $p_i$ describing the $6N$-dimensional phase space of an $N$-particle system in equilibrium, and angle brackets $\left< \cdots \right>$ represent an ensemble average. The system's Hamiltonian, $\ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace = \ensuremath{K(\{p_i\})}\xspace + \ensuremath{V(\{q_i\})}\xspace$, consists of the potential energy $\ensuremath{K(\{p_i\})}\xspace = \sum_{i=1}^{3N} p_i^2/(2m_i)$, where $m_i$ is the mass associated with the $i$-th degree of freedom, and the conservative $N$-body potential $V(\{q_i\})$. The most noteworthy feature of Eq. (1) is that ${\mathbf B}({\mathbf \Gamma})$ can be any continuous and differentiable vector in phase space (4). Choosing ${\mathbf B}({\mathbf \Gamma})= \{0, \cdots, 0, p_1, \cdots, p_{3N}\}$ yields the familiar equipartition theorem,

$$k_B T = \left< \frac{1}{N} \, \sum_{j=1}^N \frac{p_j^2}{m_j} \right>$$ (2)

where $m_j$ is the mass of the $j$-th particle. In what follows, we will use the index $j$ to label the $N$ particles and $i$ for the system's $3N$ degrees of freedom. Choosing instead ${\mathbf B}({\mathbf \Gamma})= -\nabla \ensuremath{V(\{q_i\})}\xspace$ yields

$$k_B T_{config} = \frac{\left< \nabla V\cdot \nabla V \right>}{ \left< \nabla^2 V \right>},$$ (3)

which depends only on the objects' positions and not their momenta. This has come to be called the configurational temperature.

The configurational temperature's independence of the particles' momenta has important ramifications for experimental studies of systems such as colloidal suspensions whose configurations are easily measured but whose momenta are not. The experiments described in this article exploit properties of the configurational temperature to obtain new insights into the interactions between charge stabilized colloidal particles. When combined with thermodynamic sum rules, this formalism provides previously lacking thermodynamic self-consistency tests for measurements of the particles' effective pair potentials. The same formalism also can be used to measure pair potentials in soft-matter systems directly, thereby bypassing questions of interpretation raised in previous studies, and yielding comparably accurate results with substantially less data.

Section 2 provides an overview of several consequences of Eq. (1). These are extended in Sec. 3 to a hierarchy of hyperconfigurational temperatures that lend themselves particularly nicely to experimental studies. Section 4 provides an alternate foundation for the entire configurational temperature formalism in the classical hypervirial theorem. Consistency among the myriad temperature definitions is possible only if the assumptions underlying their derivations all are satisfied simultaneously. Applying these definitions to experimental data, as described in Sec. 5, therefore probes the nature of the system's inter-particle interactions. These data were obtained from digital video microscopy measurements on monolayers of charge-stabilized colloidal spheres dispersed in water between parallel glass surfaces. Not only does this system lend itself naturally to computing the configurational temperature, but the results also help to resolve a long-standing controversy regarding the nature of charged colloids' interactions in confined geometries. These results are summarized in Sec. 6.