Governing Eq. for Thermodynamic Computing

Following my previous post on thermodynamic computing, I'm going to just write down my summary notes from section II A.1 in ref. [1]. For the most part it is just a rephrasing of the paper, so you could easily just read it, but for me I needed to write it down somewhere. There also is a more through derivation and review of Langevin dynamics in ref. [2] as well as the Fokker-Planck equations.

Langevin Equations

From my perspective the primary focus of thermodynamic computing is the use of continuous variables, that is a signal is given by $x \in \mathbb{R}$. The physics of such signal is given by stochastic dynamics¹ which contains terms related to dissipation and fluctuations. The equations which govern the dynamics are:

$$ \begin{align} \mathrm{d}x &= \frac{p}{M} \mathrm{d}t \\ \mathrm{d}p &= -\nabla U(x) - \frac{\gamma}{M} \, p \,\mathrm{d}t + \mathcal{N}\left(0, \frac{2\gamma}{\beta} \, \mathbb{I} \, \mathrm{d}t\right) \label{eq:langevin} \end{align} $$

The second equation is where the stochastic dynamics comes into play and as a result we call these types of equations stochastic differential equations (SDEs). The first equation is just the kinematic equation which links the change in generalized coordinates.

In the second equation, the first term is the conserved force due to a potential, while the second and third terms correspond to a dissipative force due to dampening-frictional term and a random force due to real-world noise, i.e., white noise that occurs at various time-scales.

The constants $\gamma$ and $\beta$ are positive and real valued, and correspond to the strength of the dissipative force while $\beta$ is usually the temperature-scale which modulates the intensity of the thermal/noise fluctuations. A smaller $\beta$ would mean more intense noise fluctuations. The $M$ corresponds to the mass which is a positive real scalar (or positive definite matrix). Depending on the relative scale of $\gamma$ and $M$ the Langevin equations are referred to as overdamped ($\gamma \gg M$) or underdamped ($\gamma \ll M$).

The eqs. $\ref{eq:langevin}$ correspond to a single trajectory in phase space. However, what we are typically most interested in is the time-evolution of the system's probability density function (PDF). This evolution is described by the Fokker-Planck equation, which is derived from the Langevin equations. The Fokker-Planck equation accounts for both the deterministic dynamics and the stochastic fluctuations captured by the Langevin equations, providing a comprehensive description of the system's statistical behavior over time. For more details on Fokker-Planck equations see ref. [2].

Fokker-Planck solution

The stationary solution of the Fokker-Planck equation, which describes the system's state at thermal equilibrium, is given by the Gibbs distribution:

$$ p(x) = \frac{1}{Z} e^{-\beta U(x)} $$

with $Z$, the partition function, ensuring normalization. This distribution emerges from the interplay between the deterministic forces that drive the system towards lower energy states and the stochastic forces that induce thermal fluctuations, highlighting the fundamental principle of equilibrium statistical mechanics.

The primary interest is now to determine what the exact PDF might be given a specific energy potential $U(x)$. In the case that $U(x)$ is quadratic, i.e., $U(x) = x^T A x + b^T x$ the Gibbs distribution becomes Gaussian²:

$$ \begin{equation} \label{eq:solution} x \sim \mathcal{N}(A^{-1}b, (\beta\,A)^{-1}) \end{equation} $$

It is important to note that $x$, along with corresponding³ momenta $p$, in this context is a vector of generalized coordinates that can represent various physical or analogous quantities, depending on the dimensionality and nature of the system being studied.

If we pay particular attention to eq. $\ref{eq:solution}$ we see that if we can draw enough samples of $x$ then we can determine $A^{-1}$, which is the inverse matrix from $U(x)$. Therefore we have a way using stochastic dynamics to invert a matrix assuming we can prepare and equilibrate the Gibbs distribution given $U(x)$. The excitement is obviously that many problems in ML/AI and science are all about inverting matrices.

Footnotes

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1. The nuanced point to note about term stochastic dynamics is it means that the dynamics are represented by a probability density function. The distinction is made here because we don't want to just say its a random variable but rather a time-evolving pdf. ↩

2. You can show that this potential indeed results in a Gaussian distribution if you use the technique to complete the square, which is just basic algebra manipulation. You'll end up with something like $p(x) = \frac{1}{(2\pi)^{n/2} (\beta^{-1} |A^{-1}|)^{1/2}} e^{-\frac{1}{2} \beta (x - \mu)^T A (x - \mu)}$, with $\mu = A^{-1}b$ and $\Sigma = (\beta\,A)^{-1}$. The assumption is $A$ is symmetric and invertible. ↩

3. This is usually referred to as the canonical conjugate momenta (see Hamiltonian mechanics). ↩

References

[1] D. Melanson, M.A. Khater, M. Aifer, K. Donatella, M.H. Gordon, T. Ahle, G. Crooks, A.J. Martinez, F. Sbahi, P.J. Coles, Thermodynamic Computing System for AI Applications, arXiv Preprint arXiv:2312.04836 (2023).

[2] L. Peliti, S. Pigolotti, Stochastic Thermodynamics: An Introduction, Princeton University Press, 2021. URL.

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