The correlations among the components of the random heat flow in an incompressible fluid are [18] :

Performing the derivative, we obtain:

In case we consider these magnitudes in the same volume, in an interval of time, Equations (5) and (6) transform

Deriving inside the bracket, we obtain

Approximating inside the bracket of Equation (9), we obtain from Equations (7), (9)

Then we can write

If is the same for the three coordinates, we obtain,

or

with

where we have used, and is the thermal noise, defined by its statistical properties, namely,

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

We used the definition of the Wiener’s process (see [20] [21] ), where is the “Wiener’s increment”.

To numerically solve Equation (2) we need to perform a discretization. This is achieved by multiplying both members by and performing the integration in the interval, namely,

or

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

At the limit the mean values at the RHS of the former equation can be reemplace by the instantenous values, this means that the overline on the expressions can be omitted, while. From the developments in the appendix we need to recall here that the “Wiener’s process” is just a Gaussian stochastic process of width. Then, at each pass of the integration we have to draw and nor- malize the result properly. That is to say, if is an aleatory number, with Gaussian distribution, centered in and width 1. In MATLAB,; consequently we can write. To conclude the discretization process, Equation (19) is transformed in the corresponding Euler’s equation giving the temporal evolution of the temperature, namely,

Defining, we have

Then the temperature relaxation time, will be

From now on the averages are over the realizations of the stochastic process. A first basic quantity of interest is the average temperature current in the long-time limit (i.e, after transients due to initial conditions have died out)

where

where is the number of realizations.

We consider a fluid in a rectangular cross section nanochannel, Figure 1, with the size along the x axis (width b) and the size along the y axis (height c). The length of the channel is denoted by L. The boundary conditions are:, ,. We have considered equal increments in the three coordinates, , consequently, we have to comply with the numerical stability condition, known as CFL (Courant-Friedrichs-Lewy), which reads for our case:

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, and the corresponding spatial grid step, in our case. Next, the corresponding time step is evaluated from the CFL condition,

. In selecting the number of temporal steps the code needs to run, , we consider the temporal

interval of, enough for dying out of the transients due to initial conditions. Hence, , where. In our simulation, we have, then, , , and, , , , , , , , , the corresponding grid is points.

To numerically evaluate the steady state solution, , we proceed as follows: First we initialize our working grid by setting all the matrix components of the temperatures equal to zero,. Next, we solve simultaneously the Equations (27). Every 10 time steps we compute the difference between the actual tempera- tures and the temperature in the previous verification. The maximum of the differences

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,. Then we add the corresponding hydrodynamic fluctuation term to the temperature equation, solving this equation over temporal points and realizations of the stochastic process. Then we analyze the temporal and spatial correlations of the temperature, focusing our study in the center line temperature,.

In Figure 2, Figure 3 are shown two views of the steady solution.

In Figure 4, is shown the temperature profile along y direction for versus the z axis distance to the wall,

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

We have performed spectral analysis of the fluctuations for the central line temperature. As an example of our results, we show in Figure 5(a) the temporal autocorrelation function for the mean (over stochastic realizations) central line (CL) temperature. We observe that the correlation function extends up to 10 ps. Correspondingly, in Figure 5(b), the Fourier transform of the former autocorrelation function (the spectral density of the mean CL temperature) goes up to the two-digit GHz band.

Regarding the spatial correlation we show in Figure 6(a) the normalized spatial (z axis) autocorrelation function for the mean CL temperature. We can observe that the correlation extends over more than half length of the cell, 600 nm, correspondingly, the Fourier transform of this function, Figure 6(b), extends up to wave numbers in the 10⁶ m⁻¹ range. As a validity test of the method we verify that the expectation value obeys the heat conduction equation. The deviations are less than 0.01%. Our present results demonstrate the generic spatially long-range character of nonequilibrium fluctuations.