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 time scales, active fluctuations always dominate thermal fluctuations. However, at intermediate length and time scales, 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 time scales.


I. INTRODUCTION
Active matter systems such as propelled particles [1], molecular motors [2], active gels [3], or active droplets [4] are driven out of thermal equilibrium by chemical processes at molecular scales. In such chemically active systems, a continuous flux of matter and energy drives chemical reactions, generates mechanical forces, or induces motion of molecules and macromolecular compounds. The chemical reactions transduce chemical energy into work or movements, and also release heat into the system. This continuous supply of heat can prevent thermalization to a homogeneous temperature reflecting the non-equilibrium character of active matter.
Living cells are paradigmatic examples of active systems [5][6][7]. Active cellular processes such as cell division, cell locomotion, the expression of genes or cellular signaling processes rely on an flux of matter and energy and the availability of chemical fuels such as adenosine triphosphate (ATP) or guanosine triphosphate (GTP) which transduce chemical energy by hydrosysis to the diphosphate forms ADP and GDP. Such processes produce entropy and maintain the cell away from thermodynamic equilibrium. They also generate and dissipate heat. Under such non-equilibrium conditions, living cells also organize the formation and dissolution of proteinrich condensates. Such condensates are membrane-less compartments of distinct chemical composition that play a key role for the spatial organize of cellular biochemistry [8,9]. Recent work studying the formation and dissolution of P granules in C. elegans embryos, suggests that the physics of phase separation governed by local thermodynamic equilibrium [10], provides an appropriate description of the formation of these condensates [11]. It was proposed that local equilibrium conditions hold to a good approximation at length scales of about 100nm and at microsecond time scales despite the non-equilibrium conditions inside a cell. This raises a fundamental question for chemically active systems in general, namely whether there generally exist length and time scales for which local equilibrium applies and, if so, what determines the crossover to systems that are lacking locally well-defined thermodynamic fields.
To tackle this question, we consider active and passive heat fluctuations in a chemically active system. The active fluctuations are related to the heat input associated with stochastic chemical reaction events and are described by a stochastic field theory with Poisson white noise. Passive fluctuations are not related to chemical events but to the stochasticity in the heat transport at local equilibrium. These fluctuations are described by a stochastic field theory with Gaussian white noise. Comparing the magnitudes and the statistical properties of both types of fluctuations, in particular, the correlation function, we identify the temporal and spatial scales at which the local equilibrium hypothesis prevails. Given a characteristic time scale, we derive an analytical expression for the maximal length scale where the passive fluctuations dominate, providing an upper bound for the volume at local equilibrium. We also investigate the properties of the stochastic field theory with Poisson white noise developed to describe the active fluctuations. The scaling behavior of the active correlation functions and the characteristics of the noise spectrum are of particular interest due to the Poissonian character of the noise. Al-arXiv:2203.04574v1 [cond-mat.stat-mech] 9 Mar 2022 though living cells are used throughout this work for illustrative purposes whenever a concrete example is needed, the analysis is more general and applies to other systems where activity is generated by chemical reactions.
This work is organized as follows: in Sec. II we formulate the equations describing the temperature profile inside the active system in three spatial dimensions, define the active and passive fluctuations, and derive their main statistical properties. In Sec. III we discuss the power spectra and higher cumulants of active temperature fluctuations considering systems of spatial dimensions d = 1, 2, 3. In Sec. IV, we compare the active and passive temperature correlation obtained in Sec. II and identify the dominant contribution as a function of the time and length scales. We obtain analytical expressions for the scales where active and passive contributions are equal, providing bounds for the time and length scales with active or passive domination. Our concluding remarks are presented in Sec. V. An appendix with more details on derivations is provided at the end of the document.

II. TEMPERATURE FLUCTUATIONS IN CHEMICALLY ACTIVE SYSTEMS
A. Temperature dynamics and fluctuations In a thermodynamic system, the temperature dynamics follows from the conservation of energy. Temperature dynamics is governed by a balance equation for heat, where ρ is the mass density, c P denotes the specific heat, and j Q is the heat current density. Moreover,Q corresponds to a heat source due to the conversion, for example, of chemical or mechanical energy into heat Q. In Eq. (1) the dot over Q implies a time derivative. We are interested in heat fluctuations due to stochastic chemical events. Individual chemical events give rise to a change in reaction enthalpy h 0 , which is released as heat. With many reactions of the same type taking place at positions x i and at times t i , the heat source is given bẏ where n(t, t 0 ) is the number of events having occurred between the initial time t 0 and t. We consider, for simplicity, a Poisson distribution where chemical events occur independently with a probability of a single event at time t and position x given by λ(x, t)d 3 x dt, where λ is the rate per unit volume. The average number of events in the time interval [t 0 , t] can thus be expressed as where V is the volume of the system and the angular brackets denote an ensemble average. For a heat release of a single chemical event h 0 , the average rate of energy released per unit volume is h 0 λ. The source termQ can be further expressed aṡ where η A is a space-and time-dependent temperature noise with an average η A = 0, which we refer to as active noise. The heat current j Q in Eq. (1) is driven by temperature gradients. In addition, there can be fluctuations that stem from the stochasticity of heat transport, associated with thermal conductivity. The heat current reads where κ is the thermal conductivity. At thermal equilibrium, the noise η Q satisfies where the indices α and β denote spatial coordinates and the variance follows from a Green-Kubo relation. Here η Q describes fluctuations of heat transport. Fluctuations in heat lead to temperature fluctuations which we define and study in the following. Combining Eqs. (3) and (4) leads to an equation for the temperature fluctuations δT ≡ T −T , whereT denotes the average temperature. To linear order, temperature fluctuations evolve according to where α ≡ κ/ρc P is the thermal diffusivity and η P ≡ −∇ · η Q /ρc P is the passive temperature noise. The average temperatureT satisfies The temperature profile δT = δT A + δT P is the superposition of the two contributions δT A and δT P , which stem from the active noise η A and passive noise η P , respectively. Assuming that the cross-correlation of both noises vanishes and using constant α for simplicity, the equations governing the dynamics of the active or the passive fluctuations δT A/P can be written as In the remainder of this section, the active and passive fluctuations are investigated separately by determining the corresponding correlation functions δT A (x, t)δT A (0, 0) and δT P (x, t)δT P (0, 0) .

B. Active fluctuations
The active fluctuations of temperature δT A (x, t) are due to active processes and the associated release of energy acting as local sources of heat. Comparing Eq. (2) with Eq. (3), the noise η A is identified as which corresponds to a white Poisson noise [12][13][14], characterized by a zero mean and delta-correlated cumulants: where the subscript "c" denotes a cumulant. Note that a Poissonian-type noise with delta-correlated cumulants is ubiquitous in physical chemistry and in biophysical systems [15,16]. For a single chemical event (n = 1) corresponding to a heat source occurring at x and t , the heat kernel and is given by [17,18]: Recall that the heat kernel G(x, t|x , t ) is formally the Green's function of the heat equation and describes the propagation of heat in the system. As the solution of an initial value problem, it breaks time-reversal invariance. The active temperature fluctuations δT A (x, t) can be expressed using the heat kernel as: and formally corresponds to the field theory of a generalized Poisson noise [19,20]. Since the number of chemically active events n(t, t 0 ) is a fluctuating variable, δT A (x, t) is a stochastic field. In the following, we study the statistical properties of the active temperature fluctuations δT A (x, t). From Eqs. (8) and (10a), the averaged temperature fluctuation vanishes as expected from a white Poisson noise. In Appendix A, we calculate the moment and cumulant generating functionals for δT A , which give the m-point cumulant (m > 1): Since the heat kernel G(x, t|x , t ) can be interpreted as a propagator between the points x at time t and x at time t, the m-point cumulant in Eq. (15) is related to the probability of having all fluctuations δT A (x i , t i ) originating from a single event at position x and time t . Considering for simplicity an infinite size system, a constant rate per unit volume and the long-time limit with t 1 , t 2 t 0 , we find that the second cumulant corresponding to the two-point correlation function is given as In the limit whereas for 4α|t Note that for equal times t 1 = t 2 , the relation above is exact.
A key finding of this work is that the two-point correlation function δT A (x 1 , t 1 )δT A (x 2 , t 2 ) c of the active fluctuations arising from Poisson-distributed chemical events follows a power-law scaling. This can be interpreted as a critical behavior as there are correlations on all length and time scales. From the equal-time correlation function, when t 1 = t 2 in Eq. (18), we obtain the critical exponent η = 0. These anomalous fluctuations are a direct consequence of the white Poisson noise and seems to be a characteristic feature of a field theory with stochastic Poisson noise.

C. Passive fluctuations
Even in the absence of active processes, there are fluctuations around the equilibrium temperature [21]. In a system of finite volume, for example, the relaxation towards equilibrium leads to an uncertainty on the actual value of T (x, t) with respect to the equilibrium temperature of the system. Similarly, in the case of local equilibrium, the temperature is fixed in each volume element with a certain uncertainty. In addition, the stochasticity of heat transport lead to fluctuations in temperature with an amplitude that depends on the thermal conductivity. These fluctuations enter the heat equation through the Gaussian white noise η P . Using η P = −∇ · η Q /(ρc P ) and Eqs. (5), the passive noise satisfies Now we study the statistics of the passive fluctuations of the temperature δT P governed by Eq. (8). Due to the Gaussian character of the noise, the only non-vanishing cumulant is the two-point correlation which can be derived using Fourier transformations. Due to the independence of active and passive noise, Eq. (8) in Fourier space becomes when using the definition of the Fourier transform The passive noise in Fourier spaceη P satisfies The two-point correlation function for passive fluctuations in Fourier space reads [22,23]: and taking the inverse Fourier transforms gives Contrary to the active fluctuations, the correlation function for the fluctuations around equilibrium does not possess any power-law scaling. In particular, if the time difference |t − t | is fixed, the two-point function decays exponentially with the distance |x − x |. In the limit whereas for the equal-time correlations, we obtain

III. POWER SPECTRA AND HIGHER CUMULANTS OF ACTIVE TEMPERATURE FLUCTUATIONS
The formalism developed in Sec. II to describe the active fluctuations is a free Poissonian stochastic field theory with Poisson white noise η A given by Eq. (9) which has vanishing mean and cumulants given in Eqs. (10a) and (10b). In this section, we discuss key features of the Poisson field theory and highlight differences to a Gaussian stochastic field [24]. For the sake of generality, we consider in this section a d-dimensional space.
The first and the second cumulants are identical to those of Gaussian white noise, but higher-order cumulants, given in Eq. (10b), are non-vanishing and deltacorrelated. The latter lead to non-trivial, higher-order cumulants of the field δT A , which are known exactly in Fourier space as with the Fourier transform defined in Eq. (21). The nonvanishing higher-order cumulants in Eq. (27) characterizes the non-Gaussian character of the Poisson field theory. Power spectra We have shown in the previous section that the equaltime correlation function of active temperature fluctuations exhibits as a power-law scaling while for passive fluctuations it is a delta function. To explore this powerlaw behavior further, we investigate the power spectra of the active temperature fluctuations for systems in 1, 2, and 3 spatial dimensions. We define the spectral density S A (|x|, ω) as follows [25]: where δT is the Fourier transform of δT in frequency space. The explicit expressions of the spectral densities are given in Eqs. (B3)-(B5). In the limit of small frequencies, we find that the spectral density scales as ω d 2 −2 , see Fig. 1 and Eq. (B6). Note that for d = 2, we have S A ∼ ω −1 which is a 1/f -noise [26][27][28]. Such type of noise is typical in biophysical systems [29] and a common feature associated with Poisson shot noise [30].

IV. ACTIVE VERSUS PASSIVE FLUCTUATIONS
We are interested in length and time scales for which either active or passive fluctuations dominate. To this end, we consider temperature correlation functions which have contributions from passive and active fluctuations. At length scales for which passive fluctuations dominate, local thermodynamics equilibrium is a valid approximation. On the contrary, for length scales where active fluctuations dominate, local equilibrium condition is not satisfied.
The two-point temperature correlation function is the sum of the corresponding correlation functions related to active and passive fluctuations, Note that the correlation function C is defined here in terms of the cumulants which are identical to the second moments as the mean fluctuations vanish. According to Eqs. (16) and (24), the correlation functions C A/P (x, t) depend on x ≡ |x|. From the same equations, we find for the two-point temperature correlation function: The correlation functions C A (x, t) and C P (x, t) of passive and active fluctuations are shown on Fig. 2(a-f). Fig. 2(a-c) depict the active and passive temperature correlations as a function of the length scale x for fixed time scales t. On length scales larger than the diffusion length of passive fluctuations, i.e. x √ 4αt, the passive correlations C P (x, t) are exponentially suppressed (Eq. (24)), whereas the active C A (x, t) correlations are independent these two values are equal, such that C A (0, τ ) = C P (0, τ ), Fig. 2(b). For time scales smaller than τ , Fig. 2(a), the passive fluctuations dominate the temperature correlation function in Eq. (31) on length scales smaller than the crossover length where W −1 is the −1 branch of the Lambert W function [31]. To obtain L co , we have used that the active and passive correlations are equal on a length scale that is larger than √ 4αt and we used Eq. (18) as an approximation for the correlation function C A defined in Eq. (30). Note that the length L co (t) exists in a range of time scales t that corresponds to the domain of the Lambert function W −1 . At x = L co , active and passive correlations are equal and for length scales larger than L co , the active fluctuations dominate the two-point function of Eq. (31). For time scales larger than τ , Fig. 2(c), the active fluctuations dominate the passive contribution on all length scales. Fig. 2(d-f) show the correlation functions C P (x, t) and C A (x, t) as function of the time scale t for fixed values of x. On time scales t x 2 /4α, the active and passive correlations scale in time as C A ∼ t − 1 2 and C P ∼ t − 3 2 as shown in Eqs. (17) and (25) such that C A ( , 2 /6α) = C P ( , 2 /6α). For length scales smaller than , Fig. 2(d), the passive contribution dominates for time scales in the range t co < t < τ with where W −1 is the −1 branch of the Lambert W function. To obtain t co , we have used that the active and passive correlations are equal on a time scale that is smaller than x 2 /4α and we used Eq. (18) as an approximation for the correlation function C A defined in Eq. (30). Note that the time t co (x) exists in a range of length scales x that corresponds to the domain of the Lambert function W −1 . Furthermore t co (x) is the inverse function of L co (t) given in Eq. (33). Finally, for length scales larger than , Fig. 2(f), the active fluctuations are larger than the passive contributions for all times. The analysis of the active and passive two-point correlations allows us identifying the regions in the temporal and spatial scales in the (t, x) plane which are dominated by either active or passive fluctuations, respectively; see On large length or time scales, larger than and τ respectively, the system is dominated by active fluctuations. On the contrary, inside the region bounded by L co in the x-direction and τ in the t-direction, passive fluctuations dominate and equilibrium is a good approximation. To quantify the relative importance of the passive versus the active contribution, we consider the ratio C A /C P . In the active region, for time scales smaller than τ and for length scales larger than L co , the passive correlation vanishes exponentially with increasing x and becomes much smaller than the active contribution already for length scales of one order of magnitude larger than L co , as seen on Fig. 3(a). In the passive dominated region, C P is at least 2, 4 and 6 orders of magnitude larger than C A for length scales smaller than L co and values of t equal to τ /10 2 , τ /10 4 and τ /10 6 respectively. This observation allows identifying the region where the local-equilibrium hypothesis holds, i.e. where the passive correlation function strongly dominates over the active contribution.
For a given chemically active system, our analysis allows us identifying the length and time scales where local thermodynamic equilibrium is an appropriate description. In the following, we consider the example of biological cells. For the biophysics of P granule assembly and disassembly, it has been suggested that local thermodynamics holds on length scales of 100nm and time scales of tens of nanoseconds (Ref. [11]). Taking the hydrolysis of ATP as a prototypical chemical event in the cell, we estimate a reaction enthalpy h 0 = 10k BT . Furthermore, we useT = 289K and estimate the rate of events per unit volume as λ 2.5 · 10 4 /s(µm) 3 , such that the heat production per by unit volume and time is Q = 10 3 W/m 3 , consistent with estimates for the heat production of living tissues [32,33]. Considering a thermal diffusivity for the cell similar to water α = 1.4 · 10 −7 m 2 /s [34] and using the mass density ρ = 10 3 kg/m 3 and the specific heat c P = 4 · 10 3 J/(kg K) of water, the spatio-temporal scales dominated by passive fluctuations are inside a region, see Fig. 3(b) (grey region bounded by solid line). In particular, the passive region exists for times shorter than τ = 1.2 · 10 5 s and lengths smaller than = 0.18m respectively. In Ref. [11], it has been observed that 10 −8 s after a local metabolic event releasing heat h 0 , this heat has spread in a volume of linear dimension of about 10 −7 m. The corresponding temperature change due to the event is about 10 −5 K and, thus small. Our analysis shows that for time scales of 10 −8 s, passive fluctuations dominate up to length scales given by the crossover length L co (10 −8 s) = 4.2 × 10 −7 m. This demonstrates that our analysis shown in Fig. 3(a-b) is consistent with estimates The solid line is the limit between active and passive regions for the parameters values h0 = 10kBT withT = 289K and λ = 2.5·10 4 /s(µm) 3 . This can be compared to the corresponding regions when the strength of the noise characterized by λh 2 0 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 α = 1.4·10 −7 m 2 /s, ρ = 10 3 kg/m 3 and cP = 4 · 10 3 J/(kg K).
In Fig. 3(a), the ratio between the active and passive correlation functions, C A /C P , has been defined as a measure of the chemical activity. The ratio depends linearly on the rate of events per unit volume λ and quadratically on the amount of heat released by individuals events h 0 . Fig. 3(b) shows the region of the (t, x)-plane dominated by passive fluctuations for different values of h 2 0 λ. Taking the values h 0 and λ estimated above for the living cell as reference (solid line), we observe that increasing (decreasing) λh 2 0 decreases (increases) the area of the passive region. The decrease is mostly due to a decrease of the rightmost boundary along the t-direction which can also be seen in Eq. (32) showing that τ is inversely proportional to λh 2 0 . However, the boundary in the x-direction does not change significantly when increasing λh 2 0 , as depicted in the inset of Fig. 3(b). The reason for this be-havior is that for time scales smaller than τ , passive fluctuations always dominate on the diagonal x = √ 4αt, corresponding to the diffusion length of the passive temperature fluctuations, independently of λ and h 0 . The trend of a decreasing area where passive fluctuations dominate confirms our expectation that the higher the rate and energy released by chemical events, the greater the activity of the system.

V. DISCUSSION
Our analysis has been motivated by quantitative studies of phase separation in living cells [11]. These raise the question under what conditions local thermodynamic equilibrium approximations provide an appropriate description of biophysical processes in living cells, even though cells are operating far from thermodynamic equilibrium. Cells host numerous active processes, the paradigmatic example is the activity of many enzymes driven by the hydrolysis of ATP such as the action of molecular motors in the cell. Such activity involves chemical events that release heat and trigger chemical changes of involved biomolecules. Standard approaches to describe such biophysical processes are based on the idea that locally temperature, pressure and other thermodynamic variables are well defined and that the nonequilibrium physics arises at larger scales by a smaller number of degrees of freedom that are maintained away from equilibrium and exhibit non-equilibrium dynamic behaviors. However this raises the question of whether and at which length and time scales such local equilibrium assumptions hold. Our work shows that even in active systems driven by stochastic active events that release heat, there typically exist length-and time-scale regimes where fluctuations are dominated by thermal fluctuations which are consistent with local thermodynamic conditions at a local equilibrium. Therefore in these regimes local equilibrium is an accurate framework to describe the non-equilibrium dynamics that emerges at larger scales.
To describe non-equilibrium conditions at small scales, we focus our work on fluctuations of heat, which can be either active, i.e. when a chemical event takes place far from equilibrium or passive when they are related to local equilibrium. To characterize active heat fluctuations we have introduced a stochastic Poisson field theory. Because of the Poissonian fluctuations, higher cumulants do not vanish and can be calculated explicitly. Furthermore, we find that correlation functions exhibit powerlaw behaviors in space and time which correspond to out-of-equilibrium critical behaviors without the need to tune parameters to a critical point. In frequency space, the noise spectrum decreases as a power-law. Notably, in two spatial dimensions, we find 1/f -noise. Poisson white noise is usually studied for discrete variables [12-14, 19, 20]. Here we present a continuum theory in space and time with Poisson statistics.
The approach presented in this work also used some approximations. First, we have assumed that the chemical reaction are all the same type, and second, have neglected non-linearities in the heat conduction equation. While it is straightforward to extend our analysis to include several types of reactions, the problem of non-linear heat transfer is more challenging and arises already in the absence of activity [18]. All these limitations present however directions for future developments.
Our analysis and the results presented here can also be applied to other active systems. Examples are particles propelled by chemical fuel, transport processes that generate heat via stochastic molecular events. The description of fluctuations in such systems in general requires accounting for the coupling to mass or momentum transfer suggesting more complex behaviors of the corresponding correlation functions [1].
We have shown that a field theory with a stochastic Poisson noise provides a framework that is well-suited to describe out-of-equilibrium fluctuations. Moreover, with the inclusion of sinks, such a theory provides a new description for birth and death processes and could potentially be extended to more general reaction-diffusion problems. Another direction to investigate is going beyond the free theory by adding interactions. Such interactions might be relevant, for example, to describe a temperature gradient or could be taken into account when describing chemical reactions.
In order to evaluate the ensemble average we generalize the method of Ref. [12] to include the spatial dependence of the stochastic variable. An integration over the probability density of the individual events and an average over the Poisson distribution P (n(t, t 0 ) = n) = 1 n! e −Λ Λ n are performed. We obtain The moment generating functional is therefore The cumulant generating functional Ψ t, The mean value vanishes and the m-point cumulant, for m > 1, is where in d spatial dimension, the heat kernel G reads Let us also highlight the derivation of the 2-point cumulant for a constant rate per unit volume λ(x, t) = λ and sending the limit of spatial integration to infinity. We have assuming that x 1 = x 2 and t 1 = t 2 and using that .
The integral over d d d is a d-dimensional Gaussian integral, we obtain the change of variable s ≡ (x 1 − x 2 ) 2 /4α(t 1 + t 2 − 2t ) gives where γ(s, x) is the lower incomplete gamma function. In three spatial dimensions, we obtain using that γ(1/2, x) = √ πErf( √ x).