Isochoric specific heat in the Dual Model of Liquids.

. We continue in this paper to illustrate the implications of the Dual Model of Liquids (DML) by deriving the expression for the isochoric specific heat as function of the collective degrees of freedom available at a given temperature and analyzing its dependence on temperature. Two main tasks will be accomplished. First, we show that the expression obtained for the isochoric specific heat in the DML is in line with the experimental results. Second, the expression will be compared with the analogous one obtained in another theoretical dual model of the liquid state, the Phonon Theory of Liquid Thermodynamics. This comparison will allow to get interesting insights about the number of collective degrees of freedom available in a liquid and on the value of the isobaric thermal expansion coefficient, two quantities that are related to each other in this framework.


Introduction
In a previous paper [1] the Dual Model of Liquids (DML) has been introduced. It is a mesoscopic model that assumes a liquid be a system made up of molecules arranged in solid-like dynamic structures in continuous rearrangement, swimming in an ocean of amorphous liquid. These solid-like aggregates, that we like to call liquid particles, interact with lattice particles, the phonons, present in liquids and responsible for the propagation of elastic and thermal energy. As consequence of the interactions, the dynamic solid-like aggregates of molecules and the phonons exchange energy and momentum with each other. This interaction is described in Figure 1, where the two elementary events at the base of the DML are represented. In events of type (a), an energetic wavepacket (throughout this paper we will use the terms wave-packets, phonons, collective excitations, interchangeably) interact with a solid-like structure, the liquid particle, transferring to it energy and momentum. Events of type (b) are just the opposite, a wave-packet interact with a liquid particle of the paper, the two brackets indicate the average over a statistical ensemble of the quantity inside them).
Starting from this considerations and applying a simple kinetic model we have calculated, among others, the expressions for several macroscopic quantities in terms of elementary parameters.
Examples are the thermal conductivity, the thermal diffusivity, the specific heat, etc.. The order of magnitude of the particle and wave-packet mean free paths and in particular of the relaxation times involved in their interactions, have also been calculated [1].
The internal energy associated to the pool of wave-packets, wp T q , is represented in the DML as a fraction of the total one, T q , namely: The classical expression for T q , where  is the medium density and V C the isochoric specific heat per unit mass, is now multiplied by m ; besides, wp Ν is the number of wave-packets per unit of volume 1 and wp  their average energy. The dynamics described above occurs at high frequencies and involves only the DoF of the lattice; the parameter m that we have introduced accounts for the 1 wp 3 ratio between the number of collective DoF surviving at temperature T, and the total number of available DoF. We will return on the meaning and values of m in the course of the paper, for now it is relevant to note that 1 0   m . Figure 1a shows that the energy acquired by the liquid particle is given back to the pool of elastic excitation a time lapse later, alike in a tunnel effect, as pointed out before,  assuming also the role of a relaxation time [1]. One of the immediate consequences of this model is that thermal (and elastic) energy spread in liquids is naturally described by a propagation equation, such as the Cattaneo equation, rather than by a diffusion equation, such as the Fourier [4], the delay time introduced into the propagation equation being physically interpreted as the time taken by the thermal energy to reappear in a different place, or by the icebergs to move from a place to the next, as in Figure 1.
Very interestingly, in the last few years several papers have appeared on international journals devoted to the liquid modelling, in particular the "Phonon theory of Liquid Thermodynamics" (PLT) [5][6][7][8][9][10][11][12][13][14][15], the theory of dissipative systems with k-gap [16][17] and a theory of specific heat derived by means of an analytical form of the Vibrational Density of States of liquids based on the existence of Instantaneous Normal Modes [18][19].
The PLT, like the DML, is a theoretical model that assumes liquids as dual systems. In the PLT the energy content of a liquid is described by a solid-like Hamiltonian, modified to account for the presence in liquids of anharmonic DoF. One of the main achievements of the PLT is that the Hamiltonian provides an expression for the isochoric specific heat V C covering the solid, glassy, liquid, gas and quantum liquids states of matter. Two expressions are obtained for the specific heat, one in harmonic approximation, the other including the anharmonic contributions to the pool of energy. The theoretical expressions arrived at have been confirmed experimentally in 21 different liquids. Moreover, PLT has also provided a theoretical interpretation of the elastic, visco-elastic and viscous behaviour response of matter to external disturbances as due to the value of the product 2  , where  is the frequency of the excitation of the DoF, and  the relaxation time as defined above. It depends on whether  is larger, comparable or smaller than  1 , respectively [1,[5][6][7][8][9][10][11][12][13][14][15].
Microscopically, the gap in k-space in liquids can be related to a finite propagation length of shear waves; indeed the gap in momentum space in liquids emerges only in the transverse spectrum, while the longitudinal one remains gapless. One needs two essential ingredients for the k-gap to emerge in the wave spectrum. First, we need a wave-like component enabling wave propagation; 2 This product is normally reported as  by many authors instead of  . It is clear that there is no ambiguity among the two definitions.
4 this is represented in the DML by the wave packets. Second, we need a dissipative effect, the process that disrupts the wave continuity and dissipates it over a certain distance, thus destroying waves with long wavelengths and giving origin to the gap in k-space. This last is obviously represented by the "wave packetliquid particle" interaction, that is the source of dissipation, and works displacing the wave packets from one place, where it is absorbed by the liquid particle, to another, where it returns to the system' energy pool, alike a tunnel effect. If  introduced above is the time during which the shear stress relaxes, then  gives the shear wave propagation length (or liquid elasticity length) [1]. The paper is organized as follows. In Section 2 the expression for the isochoric specific heat V C in the frame of reference of the DML is obtained as function of the number of collective DoF available at a given temperature, m . Then V C is compared with the analogous expression arrived at in the (PLT) [5][6][7][8][9][10][11][12][13][14][15] and their similarities, insights and the implications are discussed, in particular, about the limiting values of the collective DoF, and on that of the isobaric thermal expansion coefficient of the liquid,  , two quantities that are related to each other in this framework. In Section 3 the DML is shortly summarized and compared with the PLT. Finally in the last Section some further consequences and comparisons even between the two theories are discussed.

Isochoric specific heat in DML and PLT models: similarities and insights.
Scope of this Section is first to calculate the expression for the isochoric specific heat obtained in the DML, and then to compare it with that of the PLT. The same physical quantity is indeed obtained with two different approaches, namely a thermodynamic approach for the PLT and a mesoscopic-statistical approach for the DML. Consequently the comparison between the two expressions could shed light on some parameters adopted in their derivation. This is actually the case for m , used in the DML approach, and for  , used in the PLT model. For both these quantities, some limits are deduced from the comparison of the two expressions for V C .
The first step to accomplish the task is to calculate the specific heat in the frame of DML [1].
Let us return then to Eq.(1), which can be re-written as follows: depends on the nature of the liquid, and has been introduced for the sake of simplifying the expression. wp T q in Eqs. (1) and (2)  to the total one, Let's evaluate now the quantity in square brackets at the last member. Because . As for the temperature dependence of m , the basis for the reasoning is the experimental evidence of the presence in liquids of transversal modes [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]   One of the most important results of the PLT is that of having provided for the first time an expression for the isochoric specific heat V C for liquids that was shown to be valid for the solid, glassy, liquid, gas and quantum liquids states of matter [5][6][7][8][9][10][11][12][13][14][15]. To get this expression the Authors dT  In Eqs.(7) through (9) indexes 'H' and 'A' stay for harmonic and anharmonic respectively, while T q is the total heat content of the liquid defined in Eq.(1). The remaining terms have the same meanings as those quoted in [5,7], i.e. . The result at the mesoscopic scale is that the energy propagation by means of wave-packets becomes less and less important as the temperature increases. This is in line with Eq. (6). Ultimately, we may assume that in a liquid the total specific heat V C is contributed as:

11.
( ) where with M V C we have indicated the classical contribution due to molecular interactions, and the last two members have been obtained using Eq.(9).
We are now ready to accomplish the task by comparing Eqs. (9) and (10) among them:

12.
( ) We start by deriving the expression for dT dm taking into account that 0  dT dm : 13.
( ) Because the quantities at the denominator are all positive defined, the above relation reduces to discussing the algebraic sign of the numerator. Two possibilities may be considered, namely: from which we easily get 16.
at atmospheric pressure. In these cases the assumption introduced in Eq.(18) cannot be adopted.
There are other interesting insights that can be deduced from Eq. (16-b). Remembering that it , an upper limit can be deduced for  , namely: Because usually it is also 0   , we get the following relation holding for the specific heat: 20. [see Eqs. (10) and (12) and the related validity range]. Compiling Eqs. (12) and (20), we get the following expression for the lower limit of

Dual Model of Liquids comparison with other recent liquid modelling: the same scenario seen from different perspectives.
The liquid state was initially dealt with as a sort of extension of the gaseous one, this mainly because of its property of flowing like gases, of the lacking of the capability of transmitting shear stresses, and of adapting to the shapes of their containers. However, this approach has never allowed to go very far in the formulation of theories that were able to provide a fairly general picture of the physical quantities characterizing the liquid state. Let us take for instance the specific heat, for which strong theoretical basis exist for gaseous and solid states, but not for the liquid one.
The alternative approach, to consider the liquid state in the same way as that of a solid, has illustrious predecessors, such as Debye [37][38], Brillouin [39][40], and Frenkel [41]. However, they too have never gone so far as to provide a model that went beyond an intuitive description, this probably because of the lack of experimental evidences that could guide and/or validate it.
Nevertheless, recent measurements have given credit again to the ideas of Frenkel and Brillouin. At low frequencies, i.e. large wavelengths, it is not possible to "see" the mesoscopic structure of liquids. The entire liquid oscillates, compressing and expanding under the effect of pressure waves, which travel at the speed of sound. When one investigates the behaviour of a liquid at very high frequencies, and therefore at small wavelengths, it is discovered that it has a mesoscopic structure organized by means of solid-like dynamic aggregates, although ephemeral, that are identified here in a picturesque and imaginative way with the icebergs, elsewhere simply called pseudo-crystalline structures (see for instance [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], the idea that pseudo-crystalline structures "exist and persist" in liquids on short time-and length-scales has gradually consolidated, their size and number depending on the liquid temperature (and pressure). The presence of these ephemeral dynamical structures can obviously be deduced only when high frequency measurements are carried out, typical of INS and IXS techniques, the wavelengths of the radiation involved being small enough to interfere with them. In the DML these pseudo-crystalline structures within which elastic energy is propagated by means of (quasi) harmonic waves (phonons), interact with the rest of the (amorphous) liquid through anharmonic interactions, represented by wave-packets, which arise at their border.
10 Ultimately, a quasi-elastic propagation within icebergs is transmitted to the amorphous matrix in the form of anharmonic wave-packets, and vice-versa. Unlike in crystalline solids, the anharmonicity allows a non-infinitesimal duration of the interactions, with consequent transport not only of energy but also of momentum, and furthermore exchange of energy with the internal DoF of the icebergs.  [41]. Its inverse, , is the frequency of occurrence of particle jumps. In the Frenkel model the way of distinguishing a liquid from a solid or a gas goes around the value of the relaxation time F  . If, at a given temperature, the characteristic time t of a perturbation propagating in the liquid is much shorter than F  (or the frequency f much larger than F  ), over the time interval t the medium is seen by the perturbation as a solid, because the particles have not enough time to rearrange. The dynamic response is that of a free recoil of the liquid particle after the collision with the probe. Incidentally, this figure matches the interdependence between the relaxation time and the excitation frequency introduced in the Introduction.

11
The value obtained experimentally for the speed of sound in liquid water [25][26][27]32] proves that, in the range of momentum exchange examined and during the time lapse of the relaxation time, water behaves as a rigid network of molecules, and intermolecular forces are the same as in ice. Indeed the energy of the acoustic wave generated after the collision of the probe with the target particle of the medium, increases from the viscous to the elastic regime, i.e. upon frequency increase, due to the decrease of the acoustic dissipation. The same trend holds for the speed of sound, which manifests a positive dispersion vs frequency (PSD), while the opposite is true for the viscosity. Consequently, fingerprints of relaxation phenomena typical of collective dynamics can be found by examining the trend of the liquid physical parameters with frequency [22]. Surprisingly, and this is the demonstration of their ingenious intuitions, this conclusion is exactly the same that  . If, at a given temperature, the characteristic time t of a perturbation propagating in the liquid is much shorter than F  (or the frequency f much larger than F  ), over the time interval t the medium is seen by the perturbation as a solid, the particles have no time to rearrange. The dynamic response is that of a free recoil of the liquid particle after 12 the elastic collision with the probe (neutron or X-ray photon). The emerging spectrum is that of a solid being determined by the classical Maxwell-Boltzmann distribution [24], while the speed of sound is much higher. This case is that of "fast sound", whose velocity for water of s m / 3200  obtained experimentally [25][26][27]32] is very close to the sound velocity found in crystalline ice Ih, of  4000 m/s, confirming that, in the explored range of temperature, water is in the viscoelastic regime, i.e. they combine viscous and solid-like elastic properties, as early suggested by Maxwell [42] and subsequently by Frenkel [41]. In this configuration the medium supports one longitudinal mode and two transversal modes. At time significantly longer than t the opposite is true, the system is seen as a viscous medium and it supports only one longitudinal mode. More precisely, liquids support 2 transversal and 1 longitudinal modes at low temperature; as far as the temperature increases approaching the liquid-to-gas transition, the capability of supporting shear stresses is lost, and only the longitudinal mode survives (however, the fact that flat waves are not intrinsic excitations in disordered media like liquids, inevitably leads to mixing longitudinal and transverse excitations, unlike in crystals, these type of excitations being not orthogonal). This limit provides the border line between purely diffusive motions and vibratory, or oscillatory, motions.
What described above is equivalent to assert that liquids and solids differ only quantitatively and not qualitatively [10], the border line being the frequency F  , or the relaxation time F  , as guessed by Frenkel [41]. This concept, i.e. that the distinction between liquids and solids is quantitative rather than qualitative, was introduced for the first time by Eckart [45][46]  The solution for such a system shows many interesting key-points, the more relevant for our purposes are: 1) the interaction potential is an oscillating function (see Figure 5 in [16]), that is the two scalar fields, and therefore the two interacting sub-systems, exchange energy among them: 1  and 2  14 reduce and grow over time  , respectively. This process is not dissimilar from phonon scattering in crystals due to defects or anharmonicity where a plane-wave phonon ( 1  ) decays into other phonons (represented by 2  ) [16] and acquires a finite lifetime  as a result.
2) Being the total scalar field the product of 1  and 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 packetliquid particle interaction in 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 . 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 : large  corresponds to rare transitions of the field between different potential minima.
A distinct effect of dissipation is related to a situation where the energy of the system is not changed overall, but the propagation range of a collective mode (e.g. phonon) acquires a finite range. No dissipation takes place when a plane wave propagates in a crystal where the wave is an eigenstate. However, a plane wave dissipates in systems with structural and dynamical disorder, such as liquids. Particle dynamics in liquids involves both solid-like oscillatory motion at quasiequilibrium positions and diffusive jumps into neighbouring locations [7,9,15,41,16,47]. The first allows liquids to show a solid-like parameters when experimentally investigated over mesoscopic scales, while the jumps enable liquids to flow and associate them with viscosity. However, describing this dynamics needs consideration of a non-linear interaction allowing for both oscillation and jumps activated over potential barrier of the inter-particle potential. This problem does not originate in solids and gases. The smallness of atomic displacements in solids and the weakness of interactions in gases simplify their theoretical description, as already pointed out above. Liquids do not have those simplifying features: they combine large displacements with strong interactions (liquids do not have small parameters). For this reason, liquids are believed to be not described by theoretical tools at the same level as gases and solids [48].

15
There are interesting features that associate the presence of k-gaps in liquids with the fast sound or positive sound dispersion (PSD), the increase with frequency of the measured speed of sound over its hydrodynamic value. As first noted by Frenkel, a non-zero shear modulus of liquids implies that the propagation velocity crosses over from its hydrodynamic value  , where B and G are bulk and shear moduli, respectively.
According to the above, shear modes become propagating at high k values, implying PSD at these k points. This further implies that PSD should disappear with temperature starting from small k because the k-gap increases with temperature [16].
The positions of atoms, or better, of liquid particles in fluids are of course not permanent as in solids but temporary. This is explained by the fact that the kinetic energy, k p E , of a liquid particle is comparable with the potential energy,  It is clear from above that what we usually mean as a "liquid" is not a liquid at all in the DML, but a mixture of solid-like dynamic structures and an amorphous, i.e. liquid phase in its common sense. As consequence, any liquid parameter whose magnitude is experimentally measured at An alternativeor rather complementary -expression for the liquid specific heat has emerged from another model recently proposed by Baggioli and Zaccone [19]. Starting from the solution they found for the Vibrational DoS in liquids [18], they have obtained an expression for V C due to the only contribution of the experimentally proven presence in liquids of INMs, and neglecting the contribution of the "traditional" normal modes, on which the PLT is on the contrary based. INMs are indeed not normal modes in the traditional meaning, but rather pure imagery modes due to the anharmonic interaction potential typical of liquids. As such, they are absent in solids. Their comparison with the expressions for V C provided by DML and PLT is not straightforward.

Further discussion and conclusions.
The hypotheses behind the DML, duly described in [1] and to which the reader is referred to for an in-depth study, are actually two, and both have an experimental background. The first is that experiments performed with the IXS and INS techniques [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] have made it possible to highlight that the mesoscopic structure of liquids is characterized by the (ephemeral) presence of solid-like, pseudo-crystalline structures, whose size being of a few molecular diameters and mass of few molecules, within which the elastic waves propagate as in the corresponding solid phase. The number and size of these structures varies with the temperature and pressure of the liquid. The 18 second hypothesis is a direct consequence of the first. It consists in the assumption that elastic energy and momentum in liquids propagate by means of collective oscillations, or wave-packets, in a similar way as in crystalline solids. The perturbation propagates inside the solid-like structure by means of (quasi) harmonic waves; when it crosses the boundary, they leave way to anharmonic wave-packets, responsible for the propagation among solid-like structures and with the surrounding amorphous liquid phase. This scheme and the DML may be obtained by gathering together the early pictures of liquids of Debye [37][38] Brillouin [39][40] and Frenkel [41] in the light even of what is known today from the high frequency scattering experiments. In fact, as result of the interaction described in Figure 1, being the icebergs in continuous rearrangement, molecules continuously move from an iceberg to the nearest neighbour and icebergs too jump in the liquid from one site to the nearest-neighbour one, as hypothesized by Frenkel for the cybotactic groups introduced in [41].
Here in this paper and also elsewhere [1] we have largely discussed and commented the intimate similitude and differences of the DML and PLT models, in particular they are both dual models of liquids, both consider the presence of harmonic and anharmonic oscillations, give a pivotal role to relaxation times, etc.. All these arguments bring to the obvious conclusion that the two models show an evident visual identity: in the liquid state both harmonic modes, typical of the solid state, and anharmonic modes, typical of non-crystalline structures, are simultaneously present.
The comparison of the two expressions for the specific heats provides interesting information about the degree of excitation of vibratory modes in liquids. Besides, we have also pointed out the central role of relaxation times in both models; in particular, their role has the consequence, among others, that heat propagation in liquids is described by a propagation (Cattaneo) equation instead of a diffusive equation [4].
As outlined above, and duly discussed in [1], key point of the DML is that the liquid is composed of two systems, the liquid particles and the lattice particles; they interact among them through a harmonic potential inside the iceberg, and through an anharmonic potential outside. The presence of k-gap reinforces this point and also the fact that not only longitudinal but also transversal modes are allowed in liquids. As far as the system is in thermodynamic equilibrium, the same holds for the two subsystems; when however a non-equilibrium state is imposed, even the two populations reach two distinct equilibria. The anharmonic character of the interaction allows the two populations to exchange among them not only energy but also momentum. This is another key point of the model, because the interaction represents a mechanism through which energy is dissipated. A possible application of the DML is indeed that of providing an elementary mechanism to explain the viscosity, that is a dissipative phenomenon present in liquids [49]. In a physical model for viscosity of liquids the exchange of momentum and energy among the two subsystems -liquid particles and lattice particles -should be indeed accounted for, for instance by a suitable modification of the Navier-Stokes equation by using a two-fields Lagrangian. Another interesting topic to be investigated is the transition from inviscid liquid to gas.
If the elementary wave-packet  liquid-particle interactions would be perfectly elastic, all the energy (and momentum) lost by the wave-packet in an event such that of Figure 1a would be acquired by the particle and converted into kinetic energy, only the translational DoF being involved in the scattering process. Nevertheless, being the interaction anharmonic, not only it is not instantaneous, the momentum exchanged between wave-packet and particle is proportional to the duration of the interaction, p  (i.e. to the time-length of the wave-packet) but also the energy lost by a wave-packet in an event of type a) of Figure 1 is only partially converted into kinetic energy of the particle. The remaining part is converted into internal molecular energy, exciting the corresponding DoF. This is an important point of the model, which introduces the exchange of energy with the internal vibratory DoF of the molecular cluster in a phononic model of liquid. It is possible to define an "effective heat capacity" eff C for the total heat absorbed by (or given to) a liquid particle, taking into account the two branches of DoF. eff C results to be a complex quantity whose real part is the static contribution pertaining to external translational DoF, while the imaginary part depends on that pertaining to the internal vibratory (quantized) DoF, on the frequency and on the relaxation time [50]. The pool of energy stored into the cluster will be released after a time lag R  . Depending on the OoM of such time interval, the events of Figure 1 can be considered elastic, quasi-elastic or inelastic [1,5,15,23].
One of the peculiarities of the interactions between wave packets and liquid particles is to work as a tunnel effect, by moving quantities of energy from one place to another in a well-defined time interval, the relaxation time, during which it is kept out of the heat current. The role of relaxation times is important in all the situations in which transport phenomena are influenced by a delay occurring in the microscopic transport processes. One of these cases is that of the heat transport in  experiments. An alternative way to heat a small volume of liquid could be that on hitting the liquid with a focalised high power laser beam.
A second type of experiments could be aimed at investigating the glassy and liquid-to-solid transitions. Starting from a stationary temperature gradient, both temperatures could be lowered, thus lowering also the average temperature until the liquid solidifies. Light scattering experiments performed during the non stationary phase should allow to investigate the dynamics of the system when the glassy and liquid-to-solid transitions are crossed. In particular, the local domains should be oriented following the external temperature gradient, thus allowing the increase of the correlation lengths, of sound velocity, of thermal conductivity, etc., along the preferential direction of the external temperature gradient. A difference between the same parameters when measured along the direction of the temperature gradient with respect to those measured along a different direction should be evident from the experimental data collected. This technique could in some way introduce a sort of anisotropy into the liquid matrix, becoming the solid phase anisotropic.
In all the above experimental setups one could fruitfully exploit the characteristics of a special environment such a microgravity laboratory on orbiting platforms, where the above measurements could be pursuit without the disturbing, and often overwhelming effects due to the presence of the Earth's gravity.  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 wavepacket. The particle changes velocity and the frequency of wave-packet is shifted by the amount ( ) 1 2   − . Due to its time symmetry, we assume this mechanism is the equivalent of Onsager' reciprocity law at microscopic level [1][2][3].

Figures
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) and (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, as for instance a temperature or a concentration gradient, one type of event will prevail over the other along a preferential direction.  Close-up of the first part of the wave-packetparticle interaction shown in Figure 1a, during which the phonon transfers energy and momentum to the liquid particle. wp  is the extension of the wave-packet, and p d that of the liquid particle. Once p  has elapsed and the liquid particle has travelled by p  , the particle relaxes the energy stored into internal DoF; then it travels by R  during R  (not shown in the figure above).  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 thermal diffusion of one species with respect to the other.