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small fixes, mention mixed hierarchy trunctation

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Valentin Boettcher 2022-07-13 15:25:26 +02:00
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@ -663,6 +663,15 @@ This can be achieved by making the replacements
in the previous sections, where the quantities involved are as in
\cref{sec:hops_multibath} and \cref{eq:xiproc}.
In the light of \cref{sec:general_obs} it might be an interesting
question what impact mixed bath hierarchy states have. For a cyclic
machine with long strokes, where only one bath is coupled to the
system at a time, it might be efficient to truncate the hierarchy in a
way that discards mixed bath states more readily than single bath
hierarchy states as the correlations between the baths are expected to
be small.
\section{Time Dependent Hamiltonian}
\label{sec:timedep}
The above discussion is based on the model \cref{eq:totalH} which did
@ -1092,15 +1101,16 @@ This inequality only contains quantities that can be expected to be
finite, even in the limit of infinite baths.
As in \cref{sec:ergoonebath} we now demand periodic driving, that is
\(H(t+τ) = H(t)\) for some \(τ\geq 0\). \emph{Assume} that the system
enters a periodic steady state after the time \(n_0τ\) for some
\(H(t+τ) = H(t)\) for some \(τ\geq 0\). Now we \emph{Assume} that the
system enters a periodic steady state after the time \(n_0τ\) for some
\(n_0\in\NN\) so that \(ρ_\sys((n + n_0)τ)= ρ_\sys(n_0τ)\) for all
\(n\in\NN\). This assumption is linked to the notion of a ``finite
memory'' of the baths. In the same spirit, we \emph{assume} that the
energy change of each bath
\(ΔE_{\bath^i}^\cyc =ΔE_{\bath^i}((n+1)τ)-ΔE_{\bath^i}(nτ) =
E_{\bath^i}((n+1)τ)-E_{\bath^i}(nτ)\) is constant once the system is
in the periodic steady state.
in the periodic steady state. This behavior, at least on the system
level, is suggested by the NMQSD equation \cref{{eq:multinmqsd}}.
As the system entropy does not change over a cycle
\(ΔS_\sys^\cyc ΔS_\sys(τ (n+n_0)) - ΔS_\sys(τ n_0)=S_\sys(τ (n+n_0)) - S_\sys(τ
@ -1118,18 +1128,18 @@ definition of entropy, in~\cite{Strasberg2021Aug}\footnote{In this
work, a full dynamical theory is being derived.},
\cref{eq:secondlaw_cyclic} amounts to the Clausius form of the second
law. This definition of heat is corroborated in~\cite{Esposito2015Dec}
where is shown\footnote{for fermionic baths} that a definition of heat
where it is shown\footnote{for fermionic baths} that a definition of heat
involving any nonzero fraction of the interaction energy will lead to
the internal energy (as defined by the first law) not being an exact
differential.
In contrast to~\cite{Strasberg2021Aug}, no interpretation in terms of
thermodynamical quantities is required for \cref{eq:secondlaw_cyclic}
to be useful.
Assume that the interaction Hamiltonian in \cref{eq:katoineqsys}
vanishes periodically. In the periodic steady state the system energy
does not change during a cycle, so the whole energy change amounts to
the change in bath energy. In a setting with two baths
\cref{eq:secondlaw_cyclic} implies the Carnot bound.
to be useful. Assume that the interaction Hamiltonian in
\cref{eq:katoineqsys} vanishes periodically, so that system and bath
energy expectation values can be cleanly separated. In the periodic
steady state the system energy does not change during a cycle, so the
whole energy change amounts to the change in bath energy. In a setting
with two baths \cref{eq:secondlaw_cyclic} implies the Carnot bound.
% LocalWords: ergotropy