_{1}

We consider an incompressible fluid in a rectangular nanochannel. We solve numerically the three dimensional Fourier heat equation to get the steady solution for the temperature. Then we set and solve the Langevin equation for the temperature. We have developed equations in order to determine relaxation time of the temperature fluctuations, τ
_{T} = 4.62 × 10
^{-10}s. We have performed a spectral analysis of the thermal fluctuations, with the result that temporal correlations are in the one-digit ps range, and the thermal noise excites the thermal modes in the two-digit GHz range. Also we observe long-range spatial correlation up to more than half the size of the cell, 600 nm; the wave number, q, is in the 10
^{}
<sup>6</sup>
m
<sup>-1</sup>
range. We have also determined two thermal relaxation lengths in the z direction: l
_{1} = 1.18 nm and l
_{2} = 9.86 nm.

In recent years, with the advance of nanotechnology, there is interest in the fabrication of nano-scale devices powered by [

We believe that the knowledge of temperature correlations and the relaxation of the fluctuations could be important for a better understanding of channel fluid phenomena and design.

In the present work, we consider an incompressible fluid at rest in a nanochannel, in which the transfer of energy takes place entirely by thermal conduction. In order to report the temperature fluctuations, we set and solve the Langevin equation for the temperature.

The heat flow is related to the temperature gradient by the Fourier law. However, when fluctuations are present, there also appear spontaneous energy fluxes disconnected from this gradient. The “random” contributions to the dissipative heat flux will be designed by

The equation of heat transfer is particularly simple for an incompressible fluid at rest, in which the transfer of energy takes place entirely by thermal conduction (see [

The last term is the fluctuations contribution in accordance to Equation (1). We observe, in this case, the tem- perature equation is decoupled from the density and velocity equations.

The correlations among the components of the random heat flow in an incompressible fluid are [

Performing the derivative, we obtain:

In case we consider these magnitudes in the same volume

Deriving inside the bracket, we obtain

Approximating

Then we can write

If

or

with

where we have used

i.e., the correlation time of the noise is zero.for this term. Then

We used the definition of the Wiener’s process (see [

To numerically solve Equation (2) we need to perform a discretization. This is achieved by multiplying both members by

or

where in the last term of the former equation we have used Equation (17).

At the limit

Defining

Then the temperature relaxation time, will be

From now on the averages

where

where

We consider a fluid in a rectangular cross section nanochannel,

considering the equal sign, we obtain for the ratio of time to spatial increments

Then the discretization for the temperature equation envolving fluctuations, will be

The first step of the numerical procedure is the choice of the volume

interval of

To numerically evaluate the steady state solution,

is referred to as the error. Time integration of the equations is stopped when the error is less than a tolerance defined at the beginning of the process. We have found that a tolerance tol = 10^{−9} gives reasonable results for the steady state solution. In this first part of our numerical procedure (namely, the evaluation of the steady state solution) we use deterministic equations, i.e. random noise is not considered.

After getting the steady state solution for the temperature,

In

In

is fitted with two exponential, the corresponding thermal relaxation lengths are

We have performed spectral analysis of the fluctuations for the central line temperature. As an example of our results, we show in

Regarding the spatial correlation we show in ^{6} m^{−1} range. As a validity test of the method we verify that the expectation value

These such long range correlations appear generically for a wide class of nonequilibrium states [

We wish to thank the Fundación Santander-Central-Hispano (Programa de Visitantes Distinguidos UCM) for the support provided.