Acknowledgements

other transverse, consisting of only two variables. From the equations that govern the dynamics of the variables in these sets, the corresponding hydrodynamic modes were calculated. The longitudinal modes are two acoustic, λ<sup>1</sup> and λ2, modes (6), as well as the triplet formed by the visco-heat pair λ<sup>3</sup> and λ4, modes (7), and the director λ5, mode (8). In addition, the transverse ones are given by the shear λ<sup>6</sup> and the director λ7, modes (14). We find that the influence of the temperature gradient α and the gravitational field g occurs only in the longitudinal modes, being greater its effect � <sup>10</sup>�<sup>9</sup> � � on the visco-heat pair <sup>λ</sup><sup>3</sup> and <sup>λ</sup>4. This effect is quantified by

=Rc, Eq. (9), where R is the Rayleigh number and

� �!

=Rc , R<sup>0</sup> ek

� � ,

� �!

Rc is its critical value above which convection sets in. The developed analysis corresponding to the nonconvective regime was carried out under the condition

The analytical expressions calculated for the hydrodynamic modes of a nematic in the NESS considered exhibit behaviors that are of great interest in the following particular situations. First, if the isotropic limit of the simple fluid is taken, the NLC hydrodynamic modes reduce to those in the same state out of equilibrium, modes (17)–(19), [8, 37, 38]. If R ¼ 0, that is, in the absence of the uniform temperature gradient and the constant gravitational field, our expressions are simplified and reduce to those already reported for a nematic in the state of thermodynamic equilibrium, modes (6), (8), (14), (15), and (16), [22, 31, 46]. In this case, if we also consider the limit of the simple fluid, they agree with those of this system in

equilibrium, modes (17), (19), (22), and (23), [41, 47, 48]. When R ¼ Rc, that is, at the threshold of convection, from the triplet of longitudinal λ3, λ<sup>4</sup> and λ5, the viscoheat λ<sup>3</sup> vanishes, and λ<sup>4</sup> is the sum of the thermal and shear modes, modes (12) and (13); while that of director λ<sup>5</sup> is identical to mode (8) [37, 38]. Moreover, if in this nematic threshold of convection, the limit of the simple fluid is considered, the modes of this system are recovered: one is zero, mode (24), and the other is the sum

λ<sup>3</sup> and λ4, modes (7), become propagative; in the limit of the simple fluid, under similar conditions, the corresponding modes (18) are also propagative. The latter have been predicted theoretically [8, 37, 38] and verified experimentally [43].

However, it should be mentioned that our hydrodynamic modes λ3, λ4, and λ<sup>5</sup> do not coincide with those reported in the literature for an NLC in the same NESS considered here [44, 45], which consist in one mode due to the director, another more product of the coupling of the thermal and director modes, and a shear mode. The effect of external forces α and g is only manifested in the first two modes. This triplet is reduced to the corresponding director, thermal, and shear longitudinal modes of an NLC in the state of thermodynamic equilibrium, as well as to the thermal and shear modes of a simple fluid in such state. It should be noted that from the analytical expressions of these modes, the existence of nematic propagative modes cannot be predicted; much less, in this NESS, in the simple fluid. In addition, when the threshold of convection in the nematic is considered, the director mode is canceled, another one is the sum of the thermal and director modes, and the shear mode remains unchanged; consequently, when the limit of the simple fluid is taken, they are reduced to thermal and shear modes. This last result differs completely from the already reported [37, 38] for the hydrodynamic modes of a simple fluid at the threshold of convection, where one is zero, mode (24), and the other the sum of the thermal with the shear,

is the reference value (10), our results predict that the visco-heat pair

of the thermal and shear modes, mode (25), [37, 38]. Also, if R k

means of the Rayleigh ratio R k

Non-Equilibrium Particle Dynamics

R k � �!

=Rc ≤1:

where R<sup>0</sup> k

mode (25).

160

� �!

We thank the UACM for the economic facilities granted to cover the total cost of the publication of this research work.
