We reformulate the relaxation problem in statistical mechanics by making explicit what are the \emph{operational} objects subject to relaxation: the local time statistics of the recorded signal $Z(t)$. These local time statistics are simply the estimated histograms of observations $\{Z(t_i)\}_{i=1}^M$ performed at uniformly random times $\{t_i\}_{i=1}^M$ by a clockless observer. The subject of prediction is a belief about a future fresh out-of-sample reading of a measurement outcome whose distribution is inferred from the mathematical model believed to be true. For finite bounded systems of $N\ge 1$ degrees of freedom global irreversible relaxation of predictions can occur but special initial conditions exist. The form of the predictions depends on certain loss functions whose choice is up to the particular observer. Finally, entropy is given a learning interpretation as mutual information between the observer and the unknown past of the system under consideration and, in complete generality, its stationary value depends on the information available.
The notation $\psi(x,t)$ is among the first pieces of quantum mechanics that students learn. It is also among the easiest to over-interpret. Because $x$ and $t$ occur as arguments of the same function, students may ask whether they have the same mathematical status. They may also ask whether $\psi(t)$ should require a...
M. Bakr, Zichi Zhang, Margot Stakenborg· 0 citations
We piece together textbook material to re-derive the parameter $\beta$ entering reduced density operators from quantum statistical mechanics. By re-deriving $\beta$, we show that the content of $\beta$ depends on the particular application of statistical mechanics. We focus on two different applications, namely, electr...
Ellen T. Ekstrøm, Jacob Pedersen, Ida-Marie Høyvik· 0 citations
Let $(\xi_t)_{t \geq 0}$ be a continuous-time Markov process taking values in a countable state space $E$. We consider $k$ copies of $\xi$ started from distinct positions, and write $X = (X_t)_{t \geq 0}$ for the resulting process of configurations in $E^k$. We compare two forms of interaction between the particles: th...
We construct an information theory framework in which the fundamental objects are binary sequences of length $n$, equipped with the bitwise XOR operation. The only physical observables are counts of XOR-generated symbol classes, while the exact locations of symbols are inaccessible. Averaging over those locations ultim...
Quantum Gibbs sampling algorithms generalize classical Markov chain Monte Carlo (cMCMC) methods, and prepare thermal states on quantum computers by implementing local dissipative evolution for a mixing time $t_{\rm mix}$. Since quantum measurements disrupt the prepared state, the rate of extracting unbiased information...
Daan Timmers, B. Placke, S. Parameswaran· 0 citations
We study how the response of a Bayesian posterior statistic to future observations changes as information accumulates. For a non-decreasing function $T$, define $\Pi_n^T=\E[T(\Theta)\vert \mathcal F_n]$, where $\Theta$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditionin...
Erhan Bayraktar, Yu-Qiong Wang· 0 citations
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