Heat fluctuations in chemically active systems

Chemically active systems such as living cells are maintained out of thermal equilibrium due to chemical events which generate heat and lead to active fluctuations. A key question is to understand on which time and length scales active fluctuations dominate thermal fluctuations. Here, we formulate a stochastic field theory with Poisson white noise to describe the heat fluctuations which are generated by stochastic chemical events and lead to active temperature fluctuations. We find that on large length- and timescales, active fluctuations always dominate thermal fluctuations. However, at intermediate length- and timescales, multiple crossovers exist which highlight the different characteristics of active and thermal fluctuations. Our work provides a framework to characterize fluctuations in active systems and reveals that local equilibrium holds at certain length- and timescales.

Figure 1

Power spectra $|S_{\text{A}}|$ defined in Eq. (28) normalized by ${\overline{S}}_{\text{A},d}\equiv \lambda {\left(\frac{h_0}{\alpha \rho c_P}\right)}^2{\left|\mathit{x}\right|}^{4-d}$ as function of the rescaled frequency $\omega /\overline{\omega }$ with $\overline{\omega }=\alpha /{\left|\mathit{x}\right|}^2$ for spatial dimensions $d=1,2,3$. The scaling behavior ${\omega }^{\frac{d}{2}-2}$ is indicated.

Figure 2

Normalized two-point correlation functions ${\mathcal{C}}_{\text{P}}(x,t)$ (gray) and ${\mathcal{C}}_{\text{A}}(x,t)$ (red) for passive and active fluctuations. (a–c) Temperature correlations ${\mathcal{C}}_{\text{A/P}}$ normalized by ${\mathcal{C}}_{\tau }$ where ${\mathcal{C}}_{\tau }\equiv {\mathcal{C}}_{\text{A}}(0,\tau )={\mathcal{C}}_{\text{P}}(0,\tau )$ with $\tau$ given in Eq. (32), as function of normalized distance $x/\sqrt{\alpha \tau }$ for $t={10}^{-2}\tau$ (a), $t=\tau$ (b) and $t={10}^2\tau$ (c). (d–f) Temperature correlation ${\mathcal{C}}_{\text{A/P}}$ normalized by ${\mathcal{C}}_{\ell }$ where ${\mathcal{C}}_{\ell }\equiv {\mathcal{C}}_{\text{A}}(\ell ,{\ell }^2/6\alpha )={\mathcal{C}}_{\text{P}}(\ell ,{\ell }^2/6\alpha )$ with $\ell$ given in Eq. (34), as function of normalized time $t/\left({\ell }^2{\alpha }^{-1}\right)$ for $x={10}^{-1}\ell$ (d), $x=\ell$ (e), and $x=10\ell$ (f). The background shade indicates the dominant contribution to the correlation function: active (red), passive (gray). For the vertical line on panel (e), the contributions are equal.

Figure 3

(a) Active (red) or passive (gray) dominant contribution to the two-point correlation function $\mathcal{C}$, Eq. (31), in the $(t,x)$ plane as function of the renormalized time and length scales $t/\tau$ and $x/\ell$ with $\tau$ given in Eq. (33) and $\ell$ given in Eq. (34). The solid line depicts the boundary where ${\mathcal{C}}_{\text{A}}={\mathcal{C}}_{\text{P}}$ between the active and passive dominated regions. The white dots on the solid boundary line indicate the crossover length $L_{\text{co}}\left(t\right)$ given in Eq. (33). The intensity of the colors describes the ratio ${\mathcal{C}}_{\text{A}}/{\mathcal{C}}_{\text{P}}$. (b) Regions (shaded, overlaying) where passive fluctuations dominate in the $(t,x)$ plane as a function of timescale $t$ and length scale $x$. The solid line is the limit between active and passive regions for the parameters values $h_0=10k_B\overline{T}$ with $\overline{T}=289\text{K}$ and $\lambda =2.5\times {10}^4/\text{s}{\left(\mu \text{m}\right)}^3$. This can be compared to the corresponding regions when the strength of the noise characterized by $\lambda h_0^2$ is varied by a factor ${10}^{-2}$ (dashed), ${10}^2$ (dot-dashed), or ${10}^4$ (dotted). Note that darker regions partially cover lighter ones. Other parameters are $\alpha =1.4\times {10}^{-7}{\text{m}}^2/\text{s}$, $\rho ={10}^3\text{kg}/{\text{m}}^3$ and $c_P=4\times {10}^3\text{J}/\left(\text{kg}\text{K}\right)$.