# Physical limits to sensing material properties

## TL;DR

This paper quantifies the fundamental physical limits of sensing material properties in disordered media, providing optimal measurement protocols and implications for designing microscopic sensors, including applications in medical diagnostics.

## Contribution

It introduces a theoretical framework to determine the optimal measurement protocols for sensors embedded in heterogeneous materials, revealing fundamental sensing limits.

## Key findings

- Derived the minimal fractional uncertainty in measuring elastic constants.
- Established how sensor size and material heterogeneity affect sensing precision.
- Applied the theory to estimate bounds on cellular mechanosensing accuracy.

## Abstract

Constitutive relations describe how materials respond to external stimuli such as forces. All materials respond heterogeneously at small scales, which limits what a localized sensor can discern about the global constitution of a material. In this paper, we quantify the limits of such constitutional sensing by determining the optimal measurement protocols for sensors embedded in disordered media. For an elastic medium, we find that the least fractional uncertainty with which a sensor can determine a material constant $\lambda_0$ is approximately \begin{equation*} \frac{\delta \lambda_0}{\lambda_0 } \sim \left( \frac{\Delta_{\lambda} }{ \lambda_0^2} \right)^{1/2} \left( \frac{ d }{ a } \right)^{D/2} \left( \frac{ \xi }{ a } \right)^{D/2} \end{equation*} for $a \gg d \gg \xi$, $\lambda_0 \gg \Delta_{\lambda}^{1/2}$, and $D>1$, where $a$ is the size of the sensor, $d$ is its spatial resolution, $\xi$ is the correlation length of fluctuations in the material constant, $\Delta_{\lambda}$ is the local variability of the material constant, and $D$ is the dimension of the medium. Our results reveal how one can construct microscopic devices capable of sensing near these physical limits, e.g. for medical diagnostics. We show how our theoretical framework can be applied to an experimental system by estimating a bound on the precision of cellular mechanosensing in a biopolymer network.