Achievability of thermodynamic uncertainty relations

We first derive the multidimensional TURs by large deviation theory. Our deviation follows the similar logic with Gingrich et al. (2016), which is based on Markov jump process but can also be applied for diffusion processes. We first review the original derivation since it is also important to derive the new result. Here without loss of generality, we consider a classical thermodynamic system with $N$ mesoscopic states, with transition between states $y$ and $z$ are modelled as a continuous-time Markov jump process with rate $r(y,z)$. Let the total number of edges $(y,z)_{y<z}$ with non-zero transition rates be $E\leqslant N(N-1)/2$. Thermodynamic consistency requires local detailed balance condition, i.e., $r(y,z)\neq 0\Leftrightarrow r(z,y)\neq 0$, and we assume the system is irreducible which further ensures that there exists a unique stationary probability distribution $\pi(y)$ Van Kampen (1992). For systems poised at thermo-equilibrium, for any two mesoscopic states $y$ and $z$, the empirical current $j(y,z)$—which empirically counting the number of net jumps from $y$ to $z$ in a unit time—must converges to 0 in the long time limit because of the detailed balance. For systems far from equilibrium, instead, a non-vanishing current $j(y,z)$ may exist Qian (2007). In the long time limit, such current will converges to the associated stationary current $j^{\pi}(y,z)=\pi(y)r(y,z)-\pi(z)r(z,y)$, but fluctuations always exist, which is the observable we are interested in.

We start from level 2.5 large deviation rate function for Markov jump processes. Level 2.5 large deviation theory focuses on the joint probability distribution $\mathbb{P}(p,j)$ of the empirical density $\boldsymbol{p}\in\mathbb{R}^{N}$ and empirical current $\boldsymbol{j}\in\mathbb{R}^{E}$. In the long but finite time $T$, it has an asymptotic relation $\mathbb{P}(\boldsymbol{p},\boldsymbol{j})\asymp e^{-TI(\boldsymbol{p},\boldsymbol{j})}$, where the rate function $I(\boldsymbol{p},\boldsymbol{j})$, according to Bertini et al. (2015), is equal to

and

where $\bar{j}=p(y)r(y,z)-p(z)r(z,y)$ is the average current associated with a given empirical density and $a(y,z)=2\sqrt{p(y)r(y,z)p(z)r(z,y)}$. Then we use an upper bound $\Psi_{\text{LR}}(j)\geqslant\Psi(j,\bar{j},a)$ of each edge $(y,z)$, derived from Gingrich et al. (2016):

where $\sigma^{\pi}(y,z)=j^{\pi}(y,z)\ln\frac{\pi(y)r(y,z)}{\pi(z)r(z,y)}$ is the stationary entropy production rate of edge $(y,z)$ Ge and Qian (2010); Esposito and Van den Broeck (2010). By the contraction principle Ellis (1985), we then obtain an upper bound for the rate function of the current:

Since experimentally measuring the current between only mesoscopic states may often be challenging, what we are more interested in is the generalized current, defined as a linear combination of the current of each edge:

If only one generalized current is of interested, then the rate function of this generalized current can be bounded by letting $\boldsymbol{j}^{*}=\boldsymbol{j}^{\pi}\jmath/\jmath^{\pi}$, where $\boldsymbol{j}^{\pi}\in\mathbb{R}^{E}$ is the vector of all $j^{\pi}(y,z)$. $\boldsymbol{j}^{*}$ here automatically satisfies eq. (5), therefore by plugging in $\boldsymbol{j}^{*}$ into eq. (3) we obtain the upper bound of the rate function of $\jmath$:

Let $\Sigma^{\pi}_{\text{tot}}=\sum_{y<z}\sigma^{\pi}(y,z)$ which is equal to the total entropy production rate Esposito and Van den Broeck (2010). The relative uncertainty of $\jmath$ is defined as variance normalized by mean square:

then the bound eq. (6) becomes a bound on variance by the fact $I(\jmath^{\pi})^{\prime\prime}=1/\mathrm{Var}[\jmath]$ Ellis (1985), which leads to the scalar-value thermodynamic uncertainty relation first conjectured by A.C. Barato and U. Seifert Barato and Seifert (2015), and later proven by T.R. Gingrich et al. Gingrich et al. (2016):

Now we generalized the results above to situations when multiple generalized currents are of insterest and measured simultaneously. Consider a vector of $M$ generalized currents $\boldsymbol{\jmath}\in\mathbb{R}^{M}$, and let $\jmath_{\ell}=\sum_{y<z}d_{\ell}(y,z)j(y,z)$ be the $\ell$-th one. Without loss of generality, we only consider linear independent combination, i.e., $d_{i}(y,z)$ cannot be expressed as a linear combination of the rest $d_{\ell}(y,z)$, which further constrains that $M$ must be less or equal to $E$. From these conditions there always exist a number of $E$ vectors $\boldsymbol{g}^{(y,z)}\in\mathbb{R}^{M}$ such that

(such $\boldsymbol{g}^{(y,z)}$ is the Moore-Penrose inverse, which is not unique unless $M=E$). Plug in such $\boldsymbol{j}^{*}$ into eq. (3) and yield:

Let the vector of $M$ generalized current at the stationary state be $\boldsymbol{\jmath}^{\pi}$. The covariance matrix $\Xi$ of $\boldsymbol{\jmath}$ defined as

can be obtained from the Hessian matrix of $I(\boldsymbol{\jmath})$ at $\boldsymbol{\jmath}^{\pi}$ Ellis (1985):

where $\mathrm{H}[I(\boldsymbol{\jmath})|\boldsymbol{\jmath}^{\pi}]$ is the Hessian matrix of $I(\boldsymbol{\jmath})$ evaluated at the stationary state. Combing eq. (11) and eq. (10), we can obtain a matrix inequality¹¹1For Hermitian $A$ and $B$, $A\preceq B$ means $(B-A)$ is positive semi-definite.:

where $D$ is an $E\times E$ diagonal matrix with diagonal entries $\sigma^{\pi}(y,z)/[2j^{\pi}(y,z)^{2}]$, and $G$ is an $M\times E$ matrix with columns $\boldsymbol{g}^{(y,z)}$. Eq. (12) is hard to interpret since $G$ is not uniquely chosen, therefore we multiply $(\boldsymbol{\jmath}^{\pi})^{\top}$ and $\boldsymbol{\jmath}^{\pi}$ from left and right, respectively:

We complete the derivation by noting that $(\boldsymbol{j}^{\pi})^{\top}D\boldsymbol{j}^{\pi}=\frac{1}{2}\sum_{y<z}\sigma^{\pi}(y,z)=\frac{1}{2}\Sigma^{\pi}_{\text{tot}}$:

We finally end up with the same result as Dechant has shown in the eq. (29) in Dechant (2018) for Langevin systems. Here our approach of large deviation has been shown to also apply to continuous diffusion processes Gingrich et al. (2017), providing a proof which is a natural generalization of the original work on scalar-valued relations.

The thermodynamic uncertainty relations, both scalar-valued and vector-valued, could be potentially useful to estimate a minimum entropy production rate by experimentally measuring the fluctuation of currents Li et al. (2019); Mart´ınez et al. (2019); Manikandan et al. (2019). However, such estimation is mostly meaningful if the eq. (8) or (13) are tight bounds, i.e., their minima are achievable. It has been proven that eq. (8) is indeed tightest and can be achieved within linear-response region when the generalized current is the empirical entropy production itself Gingrich et al. (2016, 2017), which is to choose the linear combination $d(y,z)$ to be the corresponding thermodynamic force:

and the empirical entropy production rate is:

However, the tightness, or achievability, is still unclear for other generalized currents. Here we show that for any two generalized currents, the scalar-valued TURs cannot be achieved simultaneously.

First note that eq. (13) can be written as

where $\boldsymbol{\epsilon}^{-1}$ is an $M$-dimensional vector:

and

is the square root of the relative uncertainty of $\ell$-th generalized current, and $\varrho$ is the correlation coefficient matrix of $\{\jmath_{1},\jmath_{2},\cdots,\jmath_{M}\}$.

Now consider we are interested in two generalized current $\jmath_{1}$ and $\jmath_{2}$, the scalar-valued relation, eq. (8), says each uncertainty or fluctuations $\epsilon_{i}$ is bounded by two over total entropy dissipation rate. Instead, the multidimensional relation eq. (16) states:

where $\rho$ is the correlation coefficient between $\jmath_{1}$ and $\jmath_{2}$. Divided by $\sqrt{\Sigma^{\pi}_{\text{tot}}}$ on both sides, it then gives a lower bound:

Comments are in order. First, eq. (19) and (20) imply that to let the scalar-valued relation achieves the minima for both generalized currents: $\epsilon_{1}^{2}\Sigma^{\pi}_{\text{tot}}=2$ and $\epsilon_{2}^{2}\Sigma^{\pi}_{\text{tot}}=2$, a necessary condition is $\rho=1$. It means that unless the two generalized currents measured are perfectly correlated, the scalar-valued TURs cannot be achieved simultaneously.

Then one crucial question may be when the generalized current correlates positively or perfectly with another. Let the two generalized currents $\jmath_{1}=\boldsymbol{a}^{\top}\boldsymbol{j}$ and $\jmath_{2}=\boldsymbol{b}^{\top}\boldsymbol{j}$, their covariance is then:

where $\Xi(\boldsymbol{j})$ is the covariance matrix of current of each edge. By Cauchy-Bunyakovsky-Schwarz inequality, $\mathrm{Cov}(\jmath_{1},\jmath_{2})$ reaches its maximum (perfectly correlated) when and only when $\boldsymbol{a}$ and $\boldsymbol{b}$ are linear dependent. Since here are only two vectors, it means $\boldsymbol{a}=\boldsymbol{b}$. On the other words, the two scalar-valued TURs for $\jmath_{1}$ and $\jmath_{2}$ only become simultaneously achievable when $\jmath_{1}$ and $\jmath_{2}$ are equivalent, which trivially degenerates to scalar-valued case. This implies for a given system, there is at most only only generalized current which can achieve the minimum of the scalar relation eq. (8). For all other generalized currents, the multidimensional TURs may serve as a tighter bound than the scalar version.

More importantly, let $\Delta_{i}=(\epsilon_{\jmath}^{2}\Sigma^{\pi}_{\text{tot}}/2-1)\geqslant 0$ be the “tightness”, then from eq. (20) we have

which gives another trade-offs between the “tightness” of two scalar-valued TURs unless the two generalized currents are carefully chosen such that their correlation $\rho=1/\sqrt{(\Delta_{1}+1)(\Delta_{2}+1)}$.

Finally we want to emphasize that for systems near equilibrium, linear-response theory states that the variance of entropy production rate $\mathrm{Var}[\Sigma_{\text{tot}}]$ indeed reaches its minimum $2\Sigma_{\text{tot}}^{\pi}$ predicted by TURs Gingrich et al. (2017); Marconi et al. (2008), therefore for any other generalized current, its fluctuation cannot achieve the minimum $2/\Sigma_{\text{tot}}^{\pi}$ predicted by the scalar-valued TURs, which brings an underestimation of the minimum entropy production rate if the entropy production rate cannot be directly measured. For systems far from equilibrium, since the fluctuations of entropy production can probably be significant, the scalar-valued TURs of other generalized currents may become tight. However, for all generalized currents, there is still at most only one of them whose fluctuation can achieve the minimum $2/\Sigma_{\text{tot}}^{\pi}$.

To conclude, we use the large deviation theory to derive the multidimensional uncertainty relations which were first proven by Dechant Dechant (2018) by a different method, Out proof comes from a natural generalization of the approach for the scalar-value uncertainty relations. Thus our derivation can be immediately generalized to finite-time fluctuations, first-passage-time, and other scenarios which similar scalar-valued relations are all derived by large deviation. Remarkably, with our results, the whole family of TURs Gingrich et al. (2017); Gingrich and Horowitz (2017); Horowitz and Gingrich (2017) can be all derived from one formalism—the large deviation theory. We speculate these work together may formulate a uniform theory of nonequilibrium fluctuations. By applying the multidimensional relations, we further demonstrate that the scalar-valued relation is tight for only one generalized current among all the possible currents in the system. Hence the multidimensional TURs may be a better candidate to infer the minimum entropy production rate experimentally.