Density and Energy Distortion in an Oscillator Chain

Abstract

The work in this thesis is focused on the thermodynamic properties of microscopic systems in low dimen- sions which are characterized by some peculiar features. In particular we try to understand how the density and the energy of the particles in an oscillator chain can change due to the presence of non equilibrium conditions. This kind of system has been studied widely and deeply: here we choose a different approach and we deal with the problem in a different way. More precisely, instead of considering the dynamics of the chain and its interaction with the reservoirs, we focus on the heat flux that travels through the chain by adding a distorsion to the Hamiltonian that describes the oscillator chain. The chain has fixed edges and equal masses for all particles and it is defined in a phase space of 2N dimensions, with N the number of particles. The work is divided into two part: in the first one we consider for the system a harmonic potential. It is known in literature that not only numerical but also exact analytical solutions already exist in this case. Indeed we use the harmonic chain as a check, to understand if our results are in agreement with the known ones and hence if our model can be consistent. To do this we try to extrapolate the properties of this physical system by computing some basic observables that characterize the chain. We start in Chapter 3 with the analytical computation for a 2-particle system: the quantities taken into account are the kinetic temperature, the variance of the space variables, the average position of the particles and the average of the distance between two particles. We prove for these observables exact solutions exist for this model but only under some constraints on the parameter space. We then study the same observables numerically in Chapter 4 for a 50-particle system by running some Monte Carlo simulations: the results were quite promising: the qualitative behavior of all the quantities considered prove to be similar to the ones known in literature. We consider also the temperature profile to check if the relation hv i 2 i = β 1 +β 2 still holds for this model: indeed the relation still stands for our model with β 1 = 1 and β 2 having a quadratic dependence on γ. In the second part of the thesis, a new potential is considered: precisely, a α-FPU potential. In this case exact analitycal solutions do not exist yet and hence in Chapter 5 we try to investigate if with this model it is possible to overcome this problem, by computing the same observables as in Chapter 3. this task is indeed not successful, even if some approximated solutions can be obtained by considering a restricted phase space. In Chapter 6 we then run the numerical simulations for the 50-particle system for three different values of k 3 , the coupling constant of the cubic term in the FPU-potential, to understand how the system is changing as we slowly move away from the harmonic case. For k 3 = 0.001 it is straightforward to notice that the behavior is similar to the harmonic case for all variables, while for k 3 = 0.002 and k 3 = 0.003 the trend prove to be more irregular: this is due to a lack of statistics since a higher number of samples is needed in the Monte Carlo simulations for higher values of k 3 . Still, the dependence of these quantities on the non equilibrium conditions, for all the three values is the same as in the harmonic case with the momentum and the variance increasing with the intensity of the distorsion. Finally we study again the temperature profile for the different values of k 3 to comapare the results with the harmonic case: we can see that also the dependence of β 2 on γ is the same, with β 1 = 1 and β 2 with a quadratic dependence on γ and increasing as we move away from the harmonic case.