# Isochoric Specific Heat in the Dual Model of Liquids

## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic representation of inelastic collisions between wave-packets and liquid particles. The event represented in (

**a**), in which an energetic wave-packet transfers energy and momentum to a liquid particle, is commuted upon time reversal into the one represented in (

**b**), where a liquid particle transfers energy and momentum to a wave-packet. The particle changes the velocity and the frequency of the wave-packet is shifted by the amount $\left({\nu}_{2}-{\nu}_{1}\right)$. Due to its time symmetry, we assume this mechanism is the equivalent of Onsager’ reciprocity law at the microscopic level [1,2,3]. In a pure isothermal liquid, energy and momentum exchanged among the icebergs are statistically equivalent, and no net effects are produced. Events of type (

**a**) will alternate with events of type (

**b**), to keep the balance of the two energy pools unaltered. Besides, the macroscopic equilibrium will ensure also the mesoscopic equilibrium; events (

**a**,

**b**) will be equally probable along any direction, to have a zero average over time and space. On the contrary, if a symmetry breaking is introduced, for instance a temperature or a concentration gradient, one type of event will prevail over the other along a preferential direction.

## 2. Isochoric Specific Heat in the DML and the PLT Models: Similarities and Insights

_{T}is the total heat content of the liquid defined in Equation (6); The remaining terms have the same meanings as those quoted in [5,7], i.e., $\alpha $ is the coefficient of isobaric thermal expansion of the liquid, $\eta $ is its viscosity, ${\tau}_{D}$ is the Debye vibration period, and ${G}_{\infty}$ is the instantaneous shear modulus. The scope of the present section is to compare the expression given by Equation (14) for the total specific heat ${C}_{V}^{PLT}$ due to collective oscillations (phonons) as calculated in the PLT with the corresponding one calculated in the DML frame as the reference, ${C}_{V}^{DML}$, obtained from Equation (8) [1]:

## 3. DML Comparison with Other Recent Liquid Models: The Same Scenario Seen from Different Perspectives

- (1)
- The interaction potential is an oscillating function (see Figure 5 in [16]), which is the two-scalar fields, and therefore the two interacting sub-systems exchange energy among them: ${\varphi}_{1}$ and ${\varphi}_{2}$ reduce and grow over time $\langle \tau \rangle $, respectively. This process is not dissimilar from phonon scattering in crystals due to defects or anharmonicity where a plane-wave phonon (${\varphi}_{1}$) decays into other phonons (represented by ${\varphi}_{2}$) [16] and acquires a finite lifetime $\langle \tau \rangle $.
- (2)
- Being the total-scalar field of the product of ${\varphi}_{1}$ and ${\varphi}_{2}$, the total energy of the composite system does not have exponential terms depending on time due to their cancellation; consequently, the total energy is a constant of motion.
- (3)
- The motion described by the solutions of the two-scalar fields or by the function represented in Figure 5 in [16] is a typical dissipative hydrodynamic motion. If the anharmonic interaction described by the Lagrangian (i.e., the wave-packet–liquid particle interaction in the DML) has a double-well (or multi-well) form, the field can move from one minimum to another (tunnel effect) in addition to oscillating in a single well.
- (4)
- This motion is analogous to diffusive particle jumps in the liquid and represents a possible origin for the viscosity. The motion is indeed a sort of hopping motion of the field via thermal activation or tunnelling between different wells with frequency ${\nu}_{F}=1/{\tau}_{F}$. It is worth noticing that the dissipation concerns the propagation of plane waves in the anharmonic field described by the two-scalar fields of the Lagrangian. The dissipation varies as ${\nu}_{F}=1/{\tau}_{F}$: large $\langle \tau \rangle $ corresponds to rare transitions of the field between different potential minima.

## 4. Further Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Close-up of the first part of the wave-packet–particle interaction shown in Figure 1a, during which the phonon transfers energy and momentum to the liquid particle. ${\Lambda}_{wp}$ is the extension of the wave-packet, and ${d}_{p}$ is the diameter of the liquid particle. Once ${\tau}_{p}$ elapses and the liquid particle travels by ${\Lambda}_{p}$, the particle relaxes the energy stored into the internal degree of freedom; then it travels by ${\Lambda}_{R}$ during ${\tau}_{R}$ (not shown in the figure above).

**Figure 3.**Icebergs of a solid lattice fluctuating and interacting within the liquid global system at equilibrium. As far as elastic (thermal) perturbations propagate within an iceberg, they behave as in solids. The propagation velocity has then the values typical of those of the solid lattice, as found by Ruocco et al. [27], which is about 3200 m/s for the case of water. The average sizes of icebergs $\langle {\Lambda}_{0}\rangle $ have been found of some nanometers. When perturbations cross the boundary between two icebergs, ${f}^{th}$ develops and energy and momentum are transmitted from one to the nearest-neighbor iceberg. This pictorial model of liquids at the mesoscopic scale, on which the DML is based, reflects also what may be deduced from experiments performed with Inelastic X-ray Scattering techniques, able to observe liquids at such scale-lengths. In a solution, solute particles may be considered as icebergs having elastic impedance different from that of the solvent. Energy and momentum exchanged between the two types of icebergs produce a net effect, resulting in the diffusion of the solute along the concentration gradient. If a temperature gradient is imposed externally, the net effect will depend on the prevailing flux of wave-packets, which will give rise to the thermal diffusion of one species with respect to the other.

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Peluso, F.
Isochoric Specific Heat in the Dual Model of Liquids. *Liquids* **2021**, *1*, 77-95.
https://doi.org/10.3390/liquids1010007

**AMA Style**

Peluso F.
Isochoric Specific Heat in the Dual Model of Liquids. *Liquids*. 2021; 1(1):77-95.
https://doi.org/10.3390/liquids1010007

**Chicago/Turabian Style**

Peluso, Fabio.
2021. "Isochoric Specific Heat in the Dual Model of Liquids" *Liquids* 1, no. 1: 77-95.
https://doi.org/10.3390/liquids1010007