| Abstract |
Energy transfer between linear oscillators, or generally speaking discrete breathers, is an intensely scrutinized research topic. Complete transfer of finite energy between two linear weakly
coupled oscillators occurs when the said oscillators are in resonance (have the same frequency).
The assumption of weak coupling is essential for the transfer to occur adiabatically. However,
the situation is fundamentally different when non-linearity is present. In that case, the frequency of oscillators is dependent on the amplitude of the motion. Therefore, complete transfer
configurations are hard to find but not absent. Preceding works[1] prove that Targeted Energy
Transfer (TET) occurs for any non-zero coupling when the detuning function, defined as the
variation of the oscillators’ energy during the transfer conserving the total action, is bounded.
In this thesis, we propose a novel computational method for revealing the non-linear configurations leading to TET. First of all, we verify the validity of our technique in the dimer
system. The latter one is analytically solvable and integrable. Specifically, we prove that the
minimization of an appropriate Loss Function using the Gradient Descent algorithm uncovers
the desired anharmonic configurations. Continuing, we proceed with the trimer system, which
is not analytically solvable. Therefore, it is addressed to a large extent through numerical methods, like in [2],[3]. In this line of work, the trimer case consists of the dimer system and an
intermediate linear layer. We use the aforementioned method to discover the optimal patterns
of this system yielding TET. In both cases, we investigate the quantum and the classical limit.
The primary outcome of this work is that our method is capable of identifying the parameters
causing complete transfer in the dimer regime. On the other hand, the situation in the trimer
system appears to be more complicated. Specifically, the desired solution is deduced under very
specific circumstances. Additionally, we verify the extreme selectivity of TET in the non-linear
case. Finally, the extreme quantum case is of special interest since TET occurs for a whole set
of parameters instead of specific values.
The thesis is organized as follows. In the first introductory chapter, we present the DNLS
Hamiltonian in the classical case. After discussing details regarding the role of each parameter,
we quantize that Hamiltonian to shift to the quantum case. Consequently, we offer a brief
examination of the gradient descent algorithm followed by an example. This latter step is the
transitional one in order to understand the particularity of the method we introduce and the
complexity of the systems under investigation. Since, for an arbitrary system, analytical results
are absent, we introduce a Loss Function that encodes the physics interpretation of the system
but simultaneously is computationally affordable. Having a clear view of the method, the results
on the dimer and the trimer regimes follow. We conclude the thesis with a discussion on possible
improvements to the method that could render it more effective.
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