# Vibrational Model of Heat Conduction in a Fluid of Hard Spheres
## 3 Vibrational Model  
The vibrational paradigm of atomic dynamics in dense liquids has been discussed by many authors over the years, see e.g., [PERSON]. [1; 49; 50; 51]. The main assumptions involved are as follows [16]: Atoms exhibit solid-like oscillations about temporary equilibrium positions corresponding to a local minimum on the system's potential energy surface [1; 51]. These positions do not form a regular lattice like in crystalline solids, but correspond to a liquid-like structure [50]. They are also not fixed and change (diffuse or drift) with time, which allows liquids to flow. However, these rearrangements occur on much longer time scales than those characterising solid-like oscillations. Effectively, one can assume that the local configurations of atoms are preserved for some time until a fluctuation in the kinetic energy allows the positions of some of these atoms to be rearranged towards a new local minimum in the multidimensional potential energy surface. This picture allows to make important approximations about the properties of atomic motion and the mechanismsof momentum and energy transport in the liquid state. For instance, the Stokes-Einstein (SE) relation without the hydrodynamic radius naturally emerges within this vibrational paradigm under a few additional assumptions [7; 13; 15; 16]. For energy transfer, separation of time scales corresponding to fast solid-like atomic oscillations and their slow drift plays a dominant role. Namely, it is reasonable to assume that a vibrating atom transports energy from its hotter to its cooler neighbors with a characteristic energy exchange rate equal to its average vibrational frequency, \(\
u=\left\langle\omega\right\rangle/2\pi\). Each atom controls the energy transfer through an area of order \(\Delta^{2}\), where \(\Delta=\rho^{-1/3}\) is the average interatomic separation. A liquid can be approximated by a quasi-layered structure with quasi-layers that are perpendicular to the temperature gradient (applied along \(x\)-axis) and are separated by the distance \(\Delta\) (see e.g., Figure 1 from Ref. [37]). The energy difference between two neighbouring layers is \((dU/dx)\Delta\). The energy flux density can be approximated as:  
\[j\simeq-\frac{\
u}{\Delta^{2}}\left(\frac{dU}{dx}\right)\Delta=-\frac{\left\langle \omega\right\rangle}{2\pi\Delta}c_{\mathrm{P}}\frac{dT}{dx}, \tag{4}\]  
where \(c_{\mathrm{P}}=(dU/dT)_{\mathrm{P}}\) is the specific heat at constant pressure and the minus sign implies that the heat flow is down the temperature gradient. On the other hand, [PERSON]'s law of the heat transfer reads:  
\[j=-\lambda\frac{dT}{dx}. \tag{5}\]  
By comparing Equations (4) and (5) we immediately obtain:  
\[\lambda=c_{\mathrm{P}}\frac{\left\langle\omega\right\rangle}{2\pi\Delta}. \tag{6}\]  
This is essentially the expression derived in Ref. [37], except that the specific heat at constant volume, \(c_{\mathrm{V}}\), appeared there. The present choice seems in general more appropriate because pressure should be constant in equilibrium. The difference is insignificant for soft spheres, because dense fluids can be considered as essentially incompressible in a wide portion of their phase diagram not too far from the freezing curve and thus \(c_{\mathrm{P}}\simeq c_{\mathrm{V}}\) holds. For example, Equation (6) with \(c_{\mathrm{P}}\to c_{\mathrm{V}}\) works very well for soft plasma-related Coulomb and Yukawa interactions and the Lennard-Jones fluid [37; 38; 39]. In such cases it is more appropriate to use \(c_{\mathrm{V}}\) for practical estimates (since \(c_{\mathrm{V}}\) is normally easier to evaluate). This is not the case, however, in the HS limit. For hard spheres \(c_{\mathrm{V}}\equiv 1.5\) and \(c_{\mathrm{P}}\) and \(c_{\mathrm{V}}\) can differ very considerably. Before we discuss heat conduction further, it should be emphasized that the vibrational picture of atomic dynamics is not particularly suitable for the HS fluid. In contrast to softer interactions, atomic motion does not exhibit a pronounced oscillatory character [52; 53; 54].