Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

TL;DR

This paper extends the thermodynamic uncertainty relation (TUR) to underdamped Langevin systems with velocity-dependent forces, providing a more general framework applicable to various physical systems.

Contribution

The authors derive a generalized TUR for underdamped Langevin dynamics with velocity-dependent forces, expanding the applicability of TUR beyond overdamped systems.

Findings

Accurately describes trade-offs in molecular refrigerators

Applies to Brownian particles in magnetic fields

Validates with active particle systems

Abstract

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. In this study, we present a more generalized TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [preprint arXiv:1901.05715]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.

Equations

$${\mathcal{Q}}_{\text{ori}}(\Theta )\equiv \frac{\text{Var}[\Theta ]}{⟨\Theta {⟩}_{\text{s}}^2}{\sigma }_{\text{s}}\mathcal{T}\ge 2k_{\text{B}},$$

$${\mathcal{Q}}_{\text{u}}(\Theta )\equiv \frac{\text{Var}[\Theta ]}{⟨\Theta {⟩}_{\text{s}}^2}{\Sigma }_{\text{u}}\ge 2k_{\text{B}},{\Sigma }_{\text{u}}\equiv \mathcal{T}(9{\sigma }_{\text{s}}+4{\rm Y})+\Omega$$

$${\dot{x}}_i=v_i,m_i{\dot{v}}_i=F_i(\text{x},\text{v})-{\gamma }_i{v}_i+{\xi }_i,$$

$${\partial }_{t}P(\text{x},\text{v},t)=-\sum _{i=1}^N({\partial }_{x_i}{J}_{x_i}+{\partial }_{v_i}{J}_{v_i}),$$

$${\dot{\tilde{x}}}_i={\tilde{v}}_i,m_i{\dot{\tilde{v}}}_i=F_i^{†}(\tilde{\text{x}},\tilde{\text{v}})-{\gamma }_i{\tilde{v}}_i+{\xi }_i,$$

$$F_i(\text{x},\text{v})=F_i^{\text{rev}}(\text{x},\text{v})+F_i^{\text{ir}}(\text{x},\text{v}),$$

$$J_{v_i}^{\text{ir}}=\frac{1}{m_i}\left[F_i^{\text{ir}}(\text{x},\text{v})-{\gamma }_i{v}_i-\frac{T_i{\gamma }_i}{m_i}{\partial }_{v_i}\right]P(\text{x},\text{v},t).$$

$$\sigma \equiv ⟨{\dot{S}}_{\text{tot}}⟩=\sum _{i=1}^N\frac{m_i^2}{T_i{\gamma }_i}\int d\text{x}\int d\text{v}\frac{(J_{v_i}^{\text{ir}}{)}^2}{P(\text{x},\text{v},t)}$$

$$\Theta [\Gamma ]={\int }_0^{\mathcal{T}}dt\mathbf{\Lambda }(\text{x}(t),\text{v}(t))\circ \text{v}(t),$$

$${\dot{x}}_i=v_i,m_i{\dot{v}}_i=H_{\theta ,i}(\text{x},\text{v})+{\xi }_i,$$

$${\mathcal{P}}_{\theta }[\Gamma ]=\mathcal{N}{P}_{\theta }({\text{x}}_0,{\text{v}}_0)\prod _{i=1}^N\exp [-{\mathcal{A}}_i[\Gamma ]],$$

$$\frac{{\text{Var}}_{\theta }[\Theta ]}{({\partial }_{\theta }⟨\Theta {⟩}_{\theta }{)}^2}\ge \frac{1}{\mathcal{I}(\theta )},$$

$$\mathcal{I}(\theta )=-⟨{\partial }_{\theta }^2\ln P_{\theta }({\text{x}}_0,{\text{v}}_0){⟩}_{\theta }+\frac{1}{2}{⟨\sum _{i=1}^N{\int }_0^{\mathcal{T}}dt\frac{({\partial }_{\theta }{H}_{\theta ,i}{)}^2}{T_i{\gamma }_i}⟩}_{\theta }.$$

$$H_{\theta ,i}(\text{x},\text{v})=$$

$$+\frac{T_i{\gamma }_i}{m_i}\left(1-(1+\theta {)}^3\right)\frac{{\partial }_{v_i}{P}^{\text{ss}}\left(\text{x},\frac{\text{v}}{1+\theta }\right)}{P^{\text{ss}}\left(\text{x},\frac{\text{v}}{1+\theta }\right)},$$

$$P_{\theta }^{\text{ss}}(\text{x},\text{v})=\frac{P^{\text{ss}}(\text{x},\frac{\text{v}}{1+\theta })}{(1+\theta {)}^N}.$$

$${\partial }_{\theta }⟨\Theta {⟩}_{\theta ,\text{s}}{|}_{\theta =0}$$

$$=\mathcal{T}\int d\text{x}\int d\text{v}{P}^{\text{ss}}(\text{x},\text{v})\text{v}\cdot (1+\text{v}\cdot {\nabla }_{\text{v}})\mathbf{\Lambda }(\text{x},\text{v})$$

$$=⟨\Theta {⟩}_{\text{s}}+⟨{\Theta }^{\prime }{⟩}_{\text{s}}$$

$$\mathcal{I}(0)=\frac{1}{2}\left[\mathcal{T}(9{\sigma }_{\text{s}}+4{{\rm Y}}_{\text{uv}})+\Omega \right],$$

$${{\rm Y}}_{\text{uv}}=$$

$$+\frac{3}{2T_i{\gamma }_i}⟨{\mathcal{F}}_i{F}_i^{\text{ir}}{⟩}_{\text{s}}+\frac{3}{T_i}⟨v_i{F}_i^{\text{ir}}{⟩}_{\text{s}}+\frac{3}{2m_i}⟨{\partial }_{v_i}{\mathcal{F}}_i{⟩}_{\text{s}}+\frac{3{\gamma }_i}{m_i}),$$

$$\Omega =$$

$${\mathcal{F}}_i=2F_i^{\text{rev}}-F_i^{\text{ir}}-v_i{\partial }_{v_i}{F}_i.$$

$${\mathcal{Q}}_{\text{uv}}(\Theta )\equiv \frac{\text{Var}[\Theta ]}{⟨\Theta {⟩}_{\text{s}}^2}{\Sigma }_{\text{uv}}\ge 2,{\Sigma }_{\text{uv}}\equiv \frac{\mathcal{T}(9{\sigma }_{\text{s}}+4{{\rm Y}}_{\text{uv}})+\Omega }{(1+⟨{\Theta }^{\prime }{⟩}_{\text{s}}/⟨\Theta {⟩}_{\text{s}}{)}^2}$$

$$\dot{x}=v,m\dot{v}=-\alpha v-\gamma v+\xi ,$$

$$P^{\text{ss}}(v)=\sqrt{\frac{m}{2\pi T^{\prime }}}\exp \left(-\frac{m}{2T^{\prime }}{v}^2\right),$$

$$F^{\text{rev}}=0,F^{\text{ir}}=-\alpha v,\mathcal{F}=2\alpha v.$$

$${\Sigma }_{\text{uv}}^{\text{mr}}=\left(4\mathcal{T}\frac{\alpha +\gamma }{m}+4\right)/(1+⟨{\Theta }^{\prime }{⟩}_{\text{s}}/⟨\Theta {⟩}_{\text{s}}{)}^2.$$

$$⟨W^{\text{mr}}{⟩}_{\text{s}}=\mathcal{T}⟨-\alpha v^2{⟩}_{\text{s}}=-\frac{\alpha \gamma T}{m(\alpha +\gamma )}\mathcal{T}.$$

$$\text{Var}[W^{\text{mr}}]=\frac{2{\alpha }^2{\gamma }^2{T}^2}{m(\alpha +\gamma {)}^3}\left(\mathcal{T}-\frac{m[1-\exp (-2(\alpha +\gamma )\mathcal{T}/m)]}{2(\alpha +\gamma )}\right).$$

$${\mathcal{Q}}_{\text{uv}}^{\text{mr}}(W^{\text{mr}})\equiv \frac{\text{Var}[W^{\text{mr}}]}{⟨W^{\text{mr}}{⟩}^2}{\Sigma }_{\text{uv}}^{\text{mr}}=2\left(1+\frac{1}{\chi }\right)\left(1-\frac{1-e^{-2\chi }}{2\chi }\right),$$

$${\mathcal{Q}}_{\text{ori}}^{\text{mr}}(W^{\text{mr}})=\frac{2{\alpha }^2}{(\alpha +\gamma {)}^2}\left(1-\frac{1-e^{-2\chi }}{2\chi }\right).$$

Full text

I Introduction

The thermodynamic uncertainty relation (TUR) is a trade-off relation between current fluctuations and entropy production (EP) Barato ; Gingrich1 . Generally, for an accumulated current $\Theta$ over a given time period $\mathcal{T}$, such as work, displacement, etc., the TUR states that the relative fluctuation multiplied by the EP is always larger than or equal to $2k_{\textrm{B}}$ in the steady state; that is, the original TUR can be written as

[TABLE]

where $\langle\cdots\rangle_{\textrm{s}}$ denotes a steady-state average, $\textrm{Var}[\Theta]=\langle\Theta^{2}\rangle_{\textrm{s}}-\langle\Theta\rangle_{\textrm{s}}^{2}$ is the variance in $\Theta$, $\sigma_{\textrm{s}}$ is the steady-state EP rate, and $k_{\textrm{B}}$ is the Boltzmann constant. Note that we use the subscript ‘ori’ to distinguish the original TUR from other modified TURs. This relation implies that it costs a large amount of EP (heat dissipation) to achieve high accuracy (low relative fluctuation) with a stochastic motion.

The TUR was first discovered in a biological network Barato . It has since been derived for a continuous-time Markov jump process over a finite time Horowitz ; Gingrich2 ; Pietzonka1 , as well as in the long-time limit Gingrich1 , and for an overdamped Langevin system Gingrich2 ; Dechant1 ; Vu2 . It has also been shown that the TUR should be modified for discrete-time Markov jump processes Proesmans , linear-response systems Macieszczak , and periodically driven systems Barato1 ; Barato2 ; Koyuk . Moreover, the TUR was utilized in Pietzonka for understanding the relations between the power, efficiency, constancy of a heat engine Benenti ; Brandner ; Allahverdyan1 ; Karel ; Campisi ; Shiraish ; Holubec1 ; Andreas2 ; JSLee ; JSLee1 .

The validity of the TUR was questioned recently for underdamped Langevin dynamics FPS . Subsequently, Vu and Hasegawa Vu1 demonstrated that the original TUR, Eq. (1), can be violated for a squared velocity current in equilibrium and for displacement of the Brownian particle in a tilted periodic potential. They derived a modified TUR for the underdamped dynamics  Vu1 :

[TABLE]

where $\Upsilon$ is the dynamical activity and $\Omega$ is a boundary term defined as in Eq. (18).

However, Eq. (2) is derived under the assumption that an external force is only position-dependent, and not velocity-dependent. Therefore, Eq. (2) cannot be applied to a system such as that studied by Chun et al. Chun , wherein a charged Brownian particle moves under a magnetic field; specifically, they showed that Eq. (1) can be violated when a magnetic field and a rotational force are applied simultaneously. To account for the effect of a Lorentz force on the TUR, a more generalized TUR is required, taking into consideration a velocity-dependent force in the underdamped dynamics. Velocity-dependent force plays a key role in many important contexts, such as molecular refrigerators (cold damping) Liang ; Cohadon ; Pinard ; Mertz ; Jourdan ; Kim , collective motions of active/passive Brownian particles with velocity-dependent interactions vicsek ; tailleur ; sevilla ; noh ; ams ; ams2 ; GC , and certain active matter dynamics mizuno2007 ; cugliandolo ; tailleur2015 ; wijland2015 ; Fodor ; Madal .

In this work, we extend the uncertainty relation, Eq.(2) so that it is applicable to underdamped Langevin systems with a general velocity-dependent force. We find that a velocity-dependent force only changes the dynamical activity term as presented in Eq. (18). We examine the applicability of our inequality to three physical systems driven by velocity-dependent forces: a cold-damping problem, a magnetic-field involved problem, and an active matter problem. From these concrete examples, we show that our inequality is valid for a system driven by a velocity-dependent force, while the original TUR Barato and that of Vu and Hasegawa’s Vu1 do not hold. We also identify several conditions allowing the lowest bound of the inequality to be attained, which was claimed to be impossible in a previous study Vu1 .

The paper is organized as follows. In Section 2, we explain our model system and derive the generalized TUR for the underdamped Langevin system with a velocity-dependent force. The main results are presented in Eq. (20). In Section 3, we calculate our inequality for three cases: cold-damping, a charged particle in a magnetic field, and an active matter system. In Section 4, we conclude the paper with a brief summary and discussion.

II Model and generalized TUR

We consider a $N$-particle underdamped Langevin system, where the $i$-th particle ($i=1,\cdots,N$) is in contact with a heat reservoir with temperature $T_{i}$. Define $x_{i}$ and $v_{i}$ as position and velocity of the $i$-th particle, respectively. A general position-and-velocity dependent force $F_{i}(\textit{{x}},\textit{{v}})$ is applied to the $i$-th particle, where $\textit{{x}}=(x_{1},\cdots,x_{N})$ and $\textit{{v}}=(v_{1},\cdots,v_{N})$. The dynamics of the $i$-th particle is described by the following equation:

[TABLE]

where $m_{i}$, $\gamma_{i}$, and $\xi_{i}$ are the mass, the damping coefficient, and the Gaussian white noise, satisfying $\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=2k_{\textrm{B}}\gamma_{i}T_{i}\delta_{ij}\delta(t-t^{\prime})$, respectively. For brevity, we set $k_{\textrm{B}}=1$ for the following discussion. If we define $P(\textit{{x}},\textit{{v}},t)$ as the probability distribution function, this dynamics can be also described by the following Fokker-Planck equation:

[TABLE]

where the probabilistic currents $J_{x_{i}}=v_{i}P(\textit{{x}},\textit{{v}},t)$ and $J_{v_{i}}=m_{i}^{-1}[-\gamma_{i}v_{i}+F_{i}(\textit{{x}},\textit{{v}})-T_{i}\gamma_{i}m_{i}^{-1}\partial_{v_{i}}]P(\textit{{x}},\textit{{v}},t)$.

Now we consider a single trajectory in the $(\textit{{x}},\textit{{v}})$ phase space of this dynamics from time $t=0$ to $t=\mathcal{T}$, which is denoted by $\Gamma\equiv[\textit{{x}}(t),\textit{{v}}(t)]_{t=0}^{t=\mathcal{T}}$. Note that $(\textit{{x}}_{0},\textit{{v}}_{0})$ is the starting point of this trajectory. The probability density observing the trajectory $\Gamma$ is denoted by $\mathcal{P}[\Gamma]$. To calculate the EP (or irreversibility), we need to define the time-reverse dynamics Udo_review . The time-reverse position and velocity variables, $\tilde{\textit{{x}}}$ and $\tilde{\textit{{v}}}$, should satisfy

[TABLE]

where the $\dagger$ operation reverses signs of all odd parameters in the time-reversal process. $\mathcal{P}^{\dagger}[\tilde{\Gamma}]$ is the probability density observing the time-reverse trajectory $\tilde{\Gamma}\equiv[\textit{{x}}(\mathcal{T}-t),-\textit{{v}}(\mathcal{T}-t)]_{t=0}^{t=\mathcal{T}}$ in the $\dagger$ dynamics of Eq. (5). Note that there is no unique way to choose odd parameters for the time-reversal process. For example, one may regard a magnetic field as an odd parameter, so change the sign of a magnetic field in the time-reverse dynamics Andreas2 ; ChunNoh . On the other hand, one may keep the sign of the magnetic field for the irreversibility spinney ; spinney1 ; lkp ; Kwon ; Yeo , where the $\dagger$ dynamics is identical to the original time-forward dynamics. Nonetheless, we will show later that the generalized TUR does not depend on the choice of odd parameters.

With a certain choice of odd parameters, we divide the force into the reversible and irreversible one as

[TABLE]

with $F_{i}^{\textrm{rev}}(\textit{{x}},\textit{{v}})={F_{i}^{\textrm{rev}}}^{\dagger}(\textit{{x}},-\textit{{v}})$, and $F_{i}^{\textrm{ir}}(\textit{{x}},\textit{{v}})=-{F_{i}^{\textrm{ir}}}^{\dagger}(\textit{{x}},-\textit{{v}})$. Then, we get the irreversible part of the velocity component of the probability current as

[TABLE]

The total EP is determined by the ratio between the two trajectory probabilities $\mathcal{P}[\Gamma]$ and $\mathcal{P}^{\dagger}(\tilde{\Gamma})$, that is, $\Delta S_{\textrm{tot}}=\ln[\mathcal{P}[\Gamma]/\mathcal{P}^{\dagger}[\tilde{\Gamma}]]$  Udo_review . In addition, the average EP rate can be written as spinney ; Kwon :

[TABLE]

The main goal of this study is finding a generalized TUR for a general current $\Theta$ which has the following form:

[TABLE]

where ${\bf\Lambda}$ is an arbitrary $N$-dimensional vector and $\circ$ denotes the Stratonovich multiplication. To go further, we take the virtual perturbation approach by Vu and Hasegawa Dechant1 ; Vu1 ; Vu2 . First, consider an auxiliary dynamics

[TABLE]

where $H_{\theta,i}$ is an auxiliary force with $H_{\theta=0,i}=F_{i}-\gamma_{i}v_{i}$. Thus, the auxiliary dynamics (10) becomes the original dynamics (3) at $\theta=0$. The detailed form of $H_{\theta,i}$ will be given later. The trajectory probability density in the auxiliary dynamics is denoted by $\mathcal{P}_{\theta}[\Gamma]$. In the Onsager-Machlup formulation Onsager , $\mathcal{P}_{\theta}[\Gamma]$ in the Ito scheme is given by

[TABLE]

where the action $\mathcal{A}_{i}[\Gamma]=\int_{0}^{\mathcal{T}}dt\frac{1}{4T_{i}\gamma_{i}}(m_{i}\dot{v}_{i}-H_{\theta,i}(\textit{{x}},\textit{{v}}))^{2}$ and $\mathcal{N}$ is the normalization constant which is independent of $\theta$.

The trajectory-ensemble average of $\Theta[\Gamma]$ in the auxiliary dynamics is $\langle\Theta\rangle_{\theta}=\int\mathcal{D}\Gamma\Theta[\Gamma]\mathcal{P}_{\theta}[\Gamma]$. With $\textrm{Var}_{\theta}[\Theta]\equiv\langle\Theta^{2}\rangle_{\theta}-\langle\Theta\rangle_{\theta}^{2}$, the Cramér-Rao inequality yields Vu2

[TABLE]

where $\mathcal{I}$ is the Fisher information given by

[TABLE]

We slightly modify the perturbation technique presented in Ref. Vu1 in order to include a velocity-dependent force by considering the auxiliary force as

[TABLE]

where $P^{\textrm{ss}}\left(\textit{{x}},\textit{{v}}\right)$ is the steady-state solution of the original dynamics (4). By using Eq. (14), it can be easily shown that the steady-state probability distribution function of the Fokker-Planck equation of the auxiliary dynamics is given by

[TABLE]

For a general ${\bf\Lambda}(\textit{{x}}(t),\textit{{v}}(t))$, $\partial_{\theta}\langle\Theta\rangle_{\theta}$ at $\theta=0$ in the steady state becomes

[TABLE]

where $\langle\Theta\rangle_{\textrm{s}}=\mathcal{T}\int d\textit{{x}}\int d\textit{{v}}{\bf\Lambda}(\textit{{x}},\textit{{v}})\cdot\textit{{v}}P^{\textrm{ss}}(\textit{{x}},\textit{{v}})$ and $\langle\Theta^{\prime}\rangle_{\textrm{s}}=\mathcal{T}\int d\textit{{x}}\int d\textit{{v}}\left[(\textit{{v}}\cdot\nabla_{\textit{{v}}}){\bf\Lambda}(\textit{{x}},\textit{{v}})\right]\cdot\textit{{v}}P^{\textrm{ss}}(\textit{{x}},\textit{{v}})$.

Now, we calculate the Fisher information $\mathcal{I}(\theta)$ at $\theta=0$. The second bulk term in Eq. (13) can be split into the term proportional to the EP and the term proportional to the dynamic activity, while the first term yields the boundary term independent of the time duration $\mathcal{T}$. After some algebra in the steady state, we arrive at

[TABLE]

where $\sigma_{\textrm{s}}$ is the steady-state EP rate given by Eq. (8), $\Upsilon_{\textrm{uv}}$ is the generalized dynamic activity, and $\Omega$ is the boundary term, which are expressed by

[TABLE]

with

[TABLE]

From Eqs. (12), (16), and (17), the generalized TUR is written as

[TABLE]

where the subscript ‘uv’ represents the ‘underdamped dynamics with a velocity dependent force’. The main difference of Eq. (20) from Eq. (2) is the form of the dynamical activity. This difference goes away without a velocity-dependent force by setting $\partial_{v_{i}}F_{i}=0$ and $F_{i}^{\textrm{ir}}=0$. It should be emphasized that $\Sigma_{\textrm{uv}}$ is independent of the choice of odd parameters, even though the EP rate $\sigma_{\textrm{s}}$ and the dynamic activity $\Upsilon$ depends on the choice, respectively. This implies that the original TUR depends on this choice. We also note that, when ${\bf\Lambda}$ has only position-dependent terms, $\langle\Theta^{\prime}\rangle_{\textrm{s}}=0$. If ${\bf\Lambda}_{i}(\textit{{x}},\textit{{v}})=\lambda_{i}(\textit{{x}})v_{i}^{d}$, $\langle\Theta^{\prime}\rangle_{\textrm{s}}/\langle\Theta\rangle_{\textrm{s}}=d$. These cases had been previously discussed in Ref. Vu1 .

III Examples

In this section, we test our generalized TUR (20) for three systems affected by velocity-dependent forces: (i) a molecular refrigerator, (ii) a Brownian particle in a magnetic field with a rotational force, and (iii) an active matter system.

III.1 Molecular refrigerator

Here, we consider a one-dimensional Brownian particle immersed in a reservoir with temperature $T$. Its position and velocity are denoted by $x$ and $v$, respectively. An external linear dissipative force $-\alpha v$ is applied to the particle. Then, the equation of motion becomes

[TABLE]

where $\gamma$ and $\xi$ are the damping coefficient and the Gaussian white noise satisfying $\langle\xi(t)\xi(t^{\prime})\rangle=2\gamma T\delta(t-t^{\prime})$, respectively. Note that $\alpha+\gamma>0$ for the stability of the dynamics. Equation (21) describes the motion of the simplest molecular refrigerator Liang ; Cohadon ; Pinard ; Mertz ; Jourdan ; Kim . The external dissipative force reduces the thermal fluctuation of the particle when $\alpha>0$. Thus, its motion is effectively the same as that of a particle in a cooler environment, which mimics a refrigerator at the particle level. This mechanism is often used for reducing thermal fluctuations of a mesoscopic system such as a suspended mirror of interferometric detectors Cohadon ; Pinard and an atomic-force-microscope (AFM) cantilever Mertz ; Jourdan .

The steady state of Eq. (21) is given by the Boltzmann distribution, that is,

[TABLE]

where $T^{\prime}=\gamma T/(\gamma+\alpha)$ is the effective temperature.

In this problem, we choose the reversible and irreversible forces as follows:

[TABLE]

Then, it is straightforward to show that $\Upsilon_{\textrm{uv}}=(\alpha+\gamma)/m$ from the fact $\langle v^{2}\rangle_{\textrm{s}}=T^{\prime}/m$. In addition, the irreversible current becomes zero from Eq. (7), thus $\sigma_{\textrm{s}}=0$. Finally, $\Omega=4$. Therefore, $\Sigma_{\textrm{uv}}$ of the molecular refrigerator becomes

[TABLE]

Note that we can choose another reversible and irreversible force as $F^{\textrm{rev}}=-\alpha v$ and $F^{\textrm{ir}}=0$, respectively Kim . In this case, the EP rate changes as $\sigma_{\textrm{s}}=\alpha^{2}/[m(\alpha+\gamma)]$ (called as entropy pumping Kim ), while $\Sigma_{\textrm{uv}}^{\textrm{mr}}$ remains unchanged.

To test the validity of Eq. (20), we choose the work current for $\Theta$ with $\Lambda(v)=-\alpha v$. It is easy to calculate the average work in the steady state as

[TABLE]

Its variance can be also calculated explicitly (see the detailed calculation in Appendix A):

[TABLE]

Combining Eqs. (25), (26), and (24) with the fact $\langle{W^{\textrm{mr}}}^{\prime}\rangle_{\textrm{s}}/\langle W^{\textrm{mr}}\rangle_{\textrm{s}}=1$, we find the TUR factor for the work as

[TABLE]

where $\chi=\mathcal{T}/\tau_{\textrm{relax}}$ is the observation time in the unit of the relaxation time $\tau_{\textrm{relax}}\equiv m/(\alpha+\gamma)$. This factor turns out to be always larger than 2, as expected (see Fig. 1). It is noteworthy to mention that the lowest bound ($\mathcal{Q}_{\textrm{uv}}\rightarrow 2$) is reachable in both the $\chi\rightarrow 0$ (short-time) and $\chi\rightarrow\infty$ (long-time) limit. Thus, our generalized TUR provides a tight bound for work fluctuations in this case.

To confirm our analysis, we performed numerical calculations by solving Eq. (21) using a generalized velocity-Verlet algorithm Ciccotti which is correct up to the second order of simulation time step $\Delta t=0.01$. We obtained $\mathcal{Q}_{\textrm{uv}}^{\textrm{mr}}(W^{\textrm{mr}})$ by averaging over $10^{7}$ sample paths starting from random initial states sampled from the steady-state distribution (22). Numerical results are shown in Figure 1, which are perfectly matched to the analytic curve (27).

We check whether the original TUR (1) is valid or not in this model. As mentioned previously, the EP rate depends on the choice of the odd parameters. With the choice of $F^{\textrm{rev}}=0$ and $F^{\textrm{ir}}=-\alpha v$, we get $\sigma_{\textrm{s}}=0$. Therefore, the original TUR factor $\mathcal{Q}_{\textrm{ori}}^{\textrm{mr}}(W^{\textrm{mr}})=0$, which clearly violates the original TUR. With the choice of $F^{\textrm{rev}}=-\alpha v$ and $F^{\textrm{ir}}=0$, we get $\sigma_{\textrm{s}}=\alpha^{2}/[m(\alpha+\gamma)]$, leading to

[TABLE]

When $\alpha>0$, $\mathcal{Q}_{\textrm{ori}}^{\textrm{mr}}(W^{\textrm{mr}})$ becomes smaller than $2$. Again, the original TUR is broken.

Furthermore, the modified TUR found by Vu and Hasegawa (2) does not hold either. Their dynamical activity is given by Vu1

[TABLE]

where $F=-\alpha v$. The EP rate is choice-dependent as discussed above. It turns out that any choice yields a negative value for $\Sigma_{\textrm{u}}$ in Eq. (2) for some parameter range of $\alpha$, which clearly demonstrates the violation of Eq. (2).

III.2 Brownian particle in a magnetic field with a rotational force

Recently, Chun et al. Chun studied the validity of the original TUR for a Brownian particle in a magnetic field. They found that the original TUR on the work current can be broken due to a magnetic field. To find a correct uncertainty relation, we should take account of the effect of a velocity-dependent force, since the magnetic field induces a Lorentz force.

To check the generalized TUR, we consider the same system studied by Chun et al. Chun . Suppose that a charged Brownian particle is trapped in a harmonic potential with stiffness $k$ and immersed in a reservoir with temperature $T$. The particle moves in a two-dimensional space, and its position and velocity are denoted by $\textit{{x}}=(x_{1},x_{2})$ and $\textit{{v}}=(v_{1},v_{2})$, respectively. The Lorentz force $(Bv_{2},-Bv_{1})$ and a nonconservative rotational force $(\kappa x_{2},-\kappa x_{1})$ are applied to the particle. Then, the equation of motion can be written as

[TABLE]

In this dynamics, the total forces are $F_{1}=Bv_{2}-kx_{1}+\kappa x_{2}$ and $F_{2}=-Bv_{1}-kx_{2}-\kappa x_{1}$. We choose the reversible part of $F_{i}$ as $F_{1}^{\textrm{rev}}=F_{1}$ and $F_{2}^{\textrm{rev}}=F_{2}$, with the irreversible parts $F_{1}^{\textrm{ir}}=0$ and $F_{2}^{\textrm{ir}}=0$. Then, we get $\mathcal{F}_{1}=2F_{1}$ and $\mathcal{F}_{2}=2F_{2}$. With this choice, the irreversible current (7) simply becomes the heat current ChunNoh ; Kwon and the EP rate is given by

[TABLE]

where $\dot{Q}_{i}=(-\gamma v_{i}+\xi_{i})\circ v_{i}$ is the heat current in the $i$-th direction. The second equality of Eq. (31) comes from the fact $\langle\dot{Q}_{i}\rangle_{\textrm{s}}=\gamma(T/m-\langle v^{2}\rangle_{\textrm{s}})$ Kwon ; Park which can be straightforwardly shown by using the Stratonovich multiplication.

For evaluating the boundary term $\Omega$, we need to know the distribution function $P^{\textrm{ss}}(\textit{{x}},\textit{{v}})$. For this purpose, we rewrite Eq. (30) in the form of a multivariate Ornstein-Uhlenbeck process as

[TABLE]

where $\textit{{z}}=(x_{1},x_{2},v_{1},v_{2})^{\textsf{T}}$, ${\bm{\eta}}=(0,0,\xi_{1},\xi_{2})^{\textsf{T}}$, and

[TABLE]

As we are interested in the steady state, we need a condition for guaranteeing a stable solution of this linear system which is given by Chun ; LeeS

[TABLE]

We define the covariance matrix $\textsf{C}\equiv\langle\textit{{z}}\textit{{z}}^{\textsf{T}}\rangle_{\textrm{s}}$ which is the solution of the matrix equation $\textsf{A}\textsf{C}+\textsf{C}\textsf{A}^{\textsf{T}}=2\textsf{D}$, where $\textsf{D}=(\gamma T/m^{2})\textrm{diag}\{0,0,1,1\}$. The solution is Gadiner ; Chun ; LeeS

[TABLE]

Then, the distribution function can be written as Chun

[TABLE]

By inserting Eq. (36) to Eq. (18), we can explicitly calculate $\Omega$. The result is

[TABLE]

Finally, using the fact $\langle x_{1}^{2}\rangle_{\textrm{s}}=\textsf{C}_{11}$, $\langle x_{2}^{2}\rangle_{\textrm{s}}=\textsf{C}_{22}$, $\langle v_{1}^{2}\rangle_{\textrm{s}}=\textsf{C}_{33}$, $\langle v_{2}^{2}\rangle_{\textrm{s}}=\textsf{C}_{44}$, $\langle x_{1}v_{2}\rangle_{\textrm{s}}=\textsf{C}_{14}$, and $\langle x_{2}v_{1}\rangle_{\textrm{s}}=\textsf{C}_{23}$, the dynamical activity can be written as

[TABLE]

Combining Eqs. (31), (37), and (38), we have consequently

[TABLE]

where the bulk term $\mathcal{S}=9\sigma_{\textrm{s}}+4\Upsilon_{\textrm{uv}}^{\textrm{mag}}$ is given as

[TABLE]

We test our generalized TUR by taking the accumulated work $W^{\textrm{mag}}$ done by the nonconservative rotational force, that is, with ${\bf\Lambda}(\textit{{x}})=(\kappa x_{2},-\kappa x_{1})$ as

[TABLE]

As ${\bf\Lambda}$ has only position-dependent terms, $\langle{W^{\textrm{mag}}}^{\prime}\rangle_{\textrm{s}}/\langle W^{\textrm{mag}}\rangle_{\textrm{s}}=0$. The long-time behavior of the relative fluctuation of the accumulated work was already calculated in Ref. Touchette ; Chun , which is

[TABLE]

where the diffusion coefficient $D_{W}^{\textrm{mag}}$ is defined as

[TABLE]

and $\kappa_{0}=\kappa/k$, $B_{0}=B/\gamma$, and $m_{0}=km/\gamma^{2}$ are dimensionless parameters. By multiplying Eqs. (39) and (42), we finally get

[TABLE]

where the subscript ‘$\infty$’ denotes that $\mathcal{Q}_{\textrm{uv}}^{\textrm{mag}}$ is evaluated in the $\mathcal{T}\rightarrow\infty$ limit.

To confirm our bound, we numerically calculate $\mathcal{Q}_{\textrm{uv},\infty}^{\textrm{mag}}(W^{\textrm{mag}})$ for many parameter sets of $(\kappa_{0},B_{0},m_{0})$ and plot them in Fig. 2. As shown in the figure, $\mathcal{Q}_{\textrm{uv},\infty}^{\textrm{mag}}(W^{\textrm{mag}})$ satisfies our bound (20). However, it seems that the lowest bound is not reachable at all in this case; the minimum value of $\mathcal{Q}_{\textrm{uv},\infty}^{\textrm{mag}}(W^{\textrm{mag}})$ is about $\sim 100$, which indicates that $\mathcal{Q}_{\textrm{uv},\infty}^{\textrm{mag}}(W^{\textrm{mag}})$ provides a very loose (so not much useful) bound in this system. For comparison, we also plot the values of the original TUR factor $\mathcal{Q}_{\textrm{ori},\infty}^{\textrm{mag}}(W^{\textrm{mag}})$ for many parameter sets, which is given by

[TABLE]

where we calculate the EP rate $\sigma_{\textrm{s}}$ as in Eq. (31). As already reported in Ref. Chun , the origianl TUR is satisfied as $\mathcal{Q}_{\textrm{ori},\infty}^{\textrm{mag}}(W^{\textrm{mag}})\geq 2$ for $B\kappa<0$, but does not hold for $B\kappa>0$.

We remark that the original TUR is valid for all parameter regions when we take the Lorentz force as the irreversible one. In this case, the EP rate includes an extra positive unconventional term $\sigma_{\textrm{s}}^{\textrm{unc}}=2B^{2}(k+\kappa B/\gamma)/(m\gamma K)$ Kwon ; LeeS . However, it is not clear at this moment that this choice may guarantee the validity of the original TUR in a more generalized model with a magetic field as well as nonlinear nonconservative forces.

III.3 Active matter

We consider an overdamped ‘self-propelled’ or ‘active’ particle moving along a one-dimensional ring. The dynamics of an active particle is sometimes described by the active Ornstein-Uhlenbeck process (AOUP) Fodor ; Madal , which is given by

[TABLE]

where $\mu$ is the mobility, $\Phi$ is a potential applied to the particle, $f_{\textrm{nc}}$ is an applied nonconservative force, and $\eta$ is a colored noise satisfying $\langle\eta(t)\eta(0)\rangle=(D/\tau)e^{-|t|/\tau}$. In the AOUP description, the evolution of $\eta$ is simply described by

[TABLE]

where $\tau$ is the persistence time and $\xi$ is a white Gaussian noise satisfying $\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$. This overdamped motion with a colored noise can be written in the form of an underdamped dynamics. By introducing the auxiliary velocity $v\equiv\dot{x}$ and mass $m\equiv\tau/\mu$, we have Fodor ; Madal

[TABLE]

where $\Psi=\Phi+\tau\partial_{t}\Phi$ , $\gamma\equiv 1/\mu$, and the auxiliary temperature by $T\equiv D/\mu$. Equation (48) describes the underdamped motion of a Brownian particle with an external force $F=-\partial_{x}\Psi-\tau v\partial_{x}^{2}\Phi+\gamma f_{\textrm{nc}}+m\dot{f}_{\textrm{nc}}$ in a reservoir with temperature $T$. As $F$ depends on velocity $v$, the effect of a velocity-dependent force should be considered for the uncertainty relation.

Here, we test our generalized TUR with a tilted periodic potential Vu1 ; Hyeon :

[TABLE]

where $A$ is the amplitude of the potential, $L$ is the length of the ring, and $n$ is an integer number. As there is no explicit time dependence in $f_{\textrm{nc}}$ and $\Phi$, $\dot{f}_{\textrm{nc}}=0$ and $\Psi=\Phi$. We choose $F^{\textrm{ir}}$ as a velocity-dependent part of $F$. Thus, we have

[TABLE]

The steady-state EP rate can be expressed as

[TABLE]

of which the detailed derivation is presented in Appendix B.

It is not possible to proceed to calculate the TUR factor analytically with the periodic potential. For simplicity, we first consider the case of $\Phi=0$, where all calculations can be done analytically, but there is no velocity-dependent force. The auxiliary underdamped equation of motion, Eq. (48), becomes

[TABLE]

Here, we take $F=\gamma f=F^{\textrm{rev}}$ and $F^{\textrm{ir}}=0$, thus $\mathcal{F}=2F$. By making a change of variables as $V=v-f$ and $X=x-ft$, Eq. (52) becomes $\dot{X}=V$ and $m\dot{V}=-\gamma V+\sqrt{2\gamma T}\xi$, which describes a Brownian motion without an external force. Thus, it is straightforward to calculate $\langle v\rangle_{\textrm{s}}=f$, $\langle x\rangle_{\textrm{s}}=f\mathcal{T}$, $\langle v^{2}\rangle_{\textrm{s}}=f^{2}+T/m$, and $\langle x^{2}\rangle_{\textrm{s}}=f^{2}\mathcal{T}^{2}+2T\mathcal{T}/\gamma$ from the facts $\langle V\rangle_{\textrm{s}}=0$, $\langle X\rangle_{\textrm{s}}=0$, $\langle V^{2}\rangle_{\textrm{s}}=T/m$, and $\langle X^{2}\rangle_{\textrm{s}}=2T\mathcal{T}/\gamma$. In addition, the EP rate is $\sigma_{\textrm{s}}=\gamma f\langle v\rangle_{\textrm{s}}/T=\gamma f^{2}/T$ and the steady-state distribution function is $P^{\textrm{ss}}(v)=\sqrt{m/(2\pi T)}\exp[-m(v-f)^{2}/(2T)]$. Using these results, $\Sigma_{\textrm{uv}}$ of the active matter becomes

[TABLE]

We consider the accumulated work $W^{\textrm{mag}}$ done by the constant force $f$, giving $\Lambda=f$ and $\langle{W^{\textrm{mag}}}^{\prime}\rangle_{\textrm{s}}=0$. It is easy to calculate its average and variance as

[TABLE]

Then, the original TUR for this active system gives

[TABLE]

which always reaches the lowest bound of the original TUR. On the other hand, our generalized TUR gives

[TABLE]

which is always larger than $2$. The lowest bound is reachable in the limit of both $\mathcal{T}$ and $f\rightarrow\infty$.

For nonzero $\Phi$, we perform numerical simulations. For convenience, we first reduce the number of parameters of the system by rescaling the units of length, time, and energy. To this end, we introduce the dimensionless variables

[TABLE]

and $\xi_{0}(t_{0})\equiv\sqrt{\tau}\xi(t)$ satisfying $\langle\xi_{0}(t_{0})\xi_{0}(t_{0}^{\prime})\rangle=\delta(t_{0}-t_{0}^{\prime})$. Equations (46) and (47) are then written in terms of the dimensionless variables and parameters as

[TABLE]

and

[TABLE]

where $\mu_{0}\equiv(n/L)^{2}\tau T\mu$, $f_{0}\equiv n\tau f/L$, and $\Phi_{0}\equiv\Phi/T=(A_{0}/2\pi)\sin(2\pi x_{0})$ with $A_{0}=LA/(nT)$. Thus, the system is characterized by the three dimensionless parameters, $\mu_{0}$, $f_{0}$, and $A_{0}$. We numerically integrate the underdamped equations, Eqs. (58) and (59), using the second-order integrator Ciccotti . Note that the boundary term $\Omega$ cannot be calculated without knowing the explicit form of the steady-state distribution $p^{\textrm{ss}}(x,v)$. Thus, we consider the infinite-time limit ($\mathcal{T}\rightarrow\infty$), where the boundary term can be neglected. We then calculate $\langle\dot{W}^{\textrm{am}}\rangle_{\textrm{s}}$ and the diffusion coefficient $D_{W}^{\textrm{am}}=\lim_{\mathcal{T}\rightarrow\infty}\textrm{Var}[W^{\textrm{am}}]/(2\mathcal{T})$ via extrapolation from $10^{5}$ different trajectories.

The results are presented in Fig. 3. As shown in the figure, $\mathcal{Q}_{\textrm{uv},\infty}^{\textrm{am}}(W^{\textrm{am}})$ satisfies the bound (20). When $A_{0}=0$, the numerical data are consistent with our analytic result (56). For all parameter spaces, it seems that $\mathcal{Q}_{\textrm{uv},\infty}^{\textrm{am}}(W^{\textrm{am}})$ is a monotonically decreasing function of $f_{0}$ and a monotonically increasing function of $\mu_{0}$ and $A_{0}$. Furthermore, as the effect of $A_{0}$ becomes negligible for large $f$ Hyeon , it is expected that all curves can be fitted by Eq. (56) and saturate to the bound $2$ in the large $f$ limit.

IV Conclusion and Discussion

In summary, we derived a generalized TUR for underdamped Langevin dynamics with a velocity-dependent force by extending the theory of Vu and Hasegawa Vu1 , and showed that a velocity-dependent force only changes the form of the dynamical activity function, compared to the case without a velocity-dependent force. In particular, our bound does not depend on the choice of reversible and irreversible forces, in sharp contrast to the original TUR Barato and the modified TUR Vu1 , where the bounds are dependent on the choice of reversible force in the presence of velocity-dependent forces.

We examine our bound in the context of three important systems pertaining to a velocity-dependent force: a molecular refrigerator, Brownian motion in a magnetic field, and active matter. In each case, we calculate the original and our TUR for the work fluctuations and show that our TUR always holds, whereas the previously known TURs do not.

In the original TUR, the equality when the TUR factor reaches the bound is attained when the distribution is Gaussian Hyeon . Thus, the lowest bound is reached on approaching the reversible limit. However, in the case of our generalized TUR, the equality is not simply attained on reaching the reversible limit. Instead, certain system-dependent specific limits are necessary to obtain the lowest bound. Moreover, it seems that the lowest bound is not achievable for some systems, e.g., a Brownian particle in a magnetic field. Thus, in these cases, our bound becomes loose.

Finally, we comment on the uncertainty relation derived from the fluctuation theorems. Hasegawa and Vu Vu3 recently presented another type of TUR, as follows:

[TABLE]

where they assume that the detailed fluctuation theorem is satisfied for $\Delta S_{\textrm{tot}}$ (EP during the time interval $\mathcal{T}$), and the current $\Theta$ is antisymmetric under time-reversal operation of a trajectory, that is, $\Theta[\Gamma]=-\Theta[\Gamma^{\dagger}]$. This TUR is known to be valid for time-discrete Markov chains and time-dependent driving with a periodic and time-symmetric protocol Proesmans ; Vu3 . In the molecular-refrigerator case, the work current is symmetric under time reversal, such that the above TUR cannot be applied. In addition, the above TUR provides an exponentially loose bound for the long-time limit, whereas our TUR constrains the relative fluctuations linearly.

Acknowledgements.

This research was supported by the NRF Grant No. 2017R1D1A1B06035497 (HP).

Appendix A Variation of work in the molecular refrigerator

The velocity $v$ and work $W^{\textrm{mr}}$ are stochastic random variables. From the equation of motion (21) and the definition of work, one can write the stochastic differential equation for the velocity and work as

[TABLE]

For simplicity, we will omit the superscript ‘mr’ for the following discussion in this section.

By using $dW^{2}/dt=2W\dot{W}$ and $d(v^{2}W)/dt=2v\circ\dot{v}W+v^{2}\dot{W}$, we have the following equations:

[TABLE]

and

[TABLE]

In the derivation of Eq. (64), we used $\langle v^{2}\rangle_{\textrm{s}}=T^{\prime}/m$ and the property of the Gaussian distribution, $\langle v^{4}\rangle_{\textrm{s}}=3\langle v^{2}\rangle_{\textrm{s}}^{2}$. By solving (64), we get

[TABLE]

Plugging the solution (65) into Eq. (63) and integrating up to $\mathcal{T}$ yields

[TABLE]

which leads to the variance of work, Eq. (26).

Appendix B The EP rate in the active matter with a tilted periodic potential

From the Onsager-Machlup formalism Onsager , the probability densities for the forward $\Gamma$ and time-reversal $\tilde{\Gamma}$ trajectories of Eq. (48) in the Stratonovich convention are given by

[TABLE]

respectively, where $\mathcal{N}$ is a normalization constant and $G$, $\tilde{G}$, $g$, $\tilde{g}$ are give by

[TABLE]

Thus, the total EP can be written as Udo_review ; Fodor

[TABLE]

where $\mathcal{G}\equiv\tau m(\dot{v}\partial_{x})(v\partial_{x})\Phi+\gamma mv\circ\dot{v}+\tau\partial_{x}\Phi(v\partial_{x})\partial_{x}\Phi+\gamma(v\partial_{x})\Phi-\gamma\tau f(v\partial_{x})\partial_{x}\Phi-\gamma^{2}fv$. As $\Phi$ has only $x$ dependence, we can use the relation $v\partial_{x}=d/dt$. Then, we can show that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

As Eqs. (70), (71), (72), and (73) are boundary terms, their steady-state averages vanish. Thus, only remaining terms in the steady state are Eqs. (74) and (75). Then, the steady-state average of $\Delta S_{\textrm{tot}}$ becomes

[TABLE]