Thermodynamic uncertainty relation in the overdamped limit with a magnetic Lorentz force

In nonequilibrium systems, the relative fluctuation of a current has a universal trade-off relation with the entropy production, called the thermodynamic uncertainty relation (TUR). For systems with broken time reversal symmetry, its violation has been reported in specific models or in the linear response regime. Here, we derive a modified version of the TUR analytically in the overdamped limit for general Langevin dynamics with a magnetic Lorentz force causing time reversal broken. Remarkably, this modified version is simply given by the conventional TUR scaled by the ratio of the reduced effective temperature of the overdamped motion to the reservoir temperature, permitting a violation of the conventional TUR. Without the Lorentz force, this ratio becomes unity and the conventional TUR is restored. We verify our results both analytically and numerically in a specific solvable system.

In nonequilibrium systems, the relative fluctuation of a current has a universal trade-off relation with the entropy production, called the thermodynamic uncertainty relation (TUR). For systems with broken time reversal symmetry, its violation has been reported in specific models or in the linear response regime. Here, we derive a modified version of the TUR analytically in the overdamped limit for general Langevin dynamics with a magnetic Lorentz force causing time reversal broken. Remarkably, this modified version is simply given by the conventional TUR scaled by the ratio of the reduced effective temperature of the overdamped motion to the reservoir temperature, permitting a violation of the conventional TUR. Without the Lorentz force, this ratio becomes unity and the conventional TUR is restored. We verify our results both analytically and numerically in a specific solvable system.

I. INTRODUCTION
The thermodynamic uncertainty relation (TUR) states that current fluctuation has a universal trade-off relation with thermodynamic cost [1][2][3]. To be precise, the product of the relative error square of an accumulated current Φ and the total entropy production ∆S tot ss in the steady state is bounded from below as where Q Φ is called the TUR factor for the current Φ, the variance Var ss [Φ] ≡ Φ 2 ss − ( Φ ss ) 2 with a steadystate average · ss , and the Boltzmann constant k B . As an example, consider a molecular motor in a directional stochastic motion. Stochastic fluctuations and heat dissipation are two key quantities needed to be minimized for ideal performance. The above TUR enlightens that both quantities can not be minimized simultaneously and there should be a trade-off between them. After the first discovery of the TUR for systems in the linear response regime and Markov processes on simple networks [1], many studies have explored its generality and applicability by investigating various systems [3,4]. Exploiting a large deviation theory or an information theory, the TUR was derived for continuous-time Markov jump processes and overdamped Langevin dynamics [5][6][7][8][9][10]. It has also been reported that the TUR is related to the efficiency bound of molecular motors [11], the power-efficiency trade-off relation for heat engines [12][13][14], the generic stochastic equation for entropy production [15,16], the Cramer-Rao inequality [9,10], and the symmetry of the joint distribution for the current and the entropy production [17,18] known as the detailed fluctuation theorem [19][20][21]. Recently, various studies have * jmpark19@kias.re.kr † hgpark@kias.re.kr developed methods for the entropy production inference based on the TUR [22][23][24][25].
Some recent studies have shown that the TUR is violated and should be modified when dynamics has an intrinsic time scale [26][27][28][29][30][31], breaks the time-reversal symmetry [32][33][34], or involves any odd-parity variable under time reversal such as velocity [35][36][37]. Especially, in the underdamped Langevin dynamics, the TUR is trivially violated for reversible currents [36] or for finite duration time [38]. Some evidence supports that the TUR could be valid for irreversible currents in the long-time limit [38], but a rigorous proof is missing. Although a variant of the TUR including dynamic activity was reported for the underdamped Langevin systems [36] and then extended to systems with velocity-dependent forces [37], their use is limited because their lower bounds depend on dynamic details and become trivial in the overdamped limit.
Recently, Chun et al [34] showed that, in an exactly solvable underdamped Langevin system with a magnetic Lorentz force, work and heat currents violate the conventional TUR of Eq. (1) even in the small-inertia (overdamped) limit. This violation may not be surprising as the Lorentz force breaks the time reversal symmetry. Instead, they reported a modified bound for the TUR factor in the overdamped limit for infinitely long duration time. Interestingly, this modified lower bound is very similar to the conventional TUR bound except for a simple dimensionless multiplication factor. However, this system is a special harmonic system, thus the applicability of their finding to general systems should not be taken for granted.
In this paper, we consider a general underdamped dynamics with a magnetic Lorentz force and rigorously show that Chun et al's finding is surprisingly still intact for general systems in the overdamped limit. More remarkably, this modified TUR is valid for general currents (odd under time reversal) including work and heat currents and even for a finite duration time. This modi-fied TUR reads where T is the reservoir temperature and T B (< T ) is the reduced effective temperature of the overdamped motion, which will be defined in the next section. We note that T B = T without a Lorentz force and the conventional TUR is restored. The derivation is rather tricky, mainly because the overdamped (small-inertia) limit in the presence of a Lorentz force generates a non-white noise [39]. We introduce an alternative but equivalent description with a white noise by employing a standard small-inertia expansion [40] on an extended Fokker-Planck equation with a current as an additional variable. The main strategy to derive the modified TUR is based on the Cramer-Rao inequality with a slightly different perturbation from the conventional one adopted in Refs. [9,10]. We also generalize the modified TUR for an arbitrary initial state. Finally, we calculate analytically the finite-time TUR factors for work, heat, and entropy production currents in the overdamped limit for the solvable harmonic system with a Lorentz force in two dimensions, which confirm our modified TUR.

II. THE OVERDAMPED LANGEVIN EQUATION WITH A MAGNETIC FIELD
For convenience, we consider two-dimensional dynamics of a charged Brownian particle in thermal contact with a heat bath at temperature T and subjected to a static magnetic field perpendicular to the motion plane. We also consider a general in-plane force f (x) acting on the particle, as shown in Fig. 1. As the Lorentz force induced by the magnetic field has an in-plane component only, the particle motion is constrained on the twodimensional plane with a proper initial condition.
The equation of motion of the particle is given by the Langevin equation aṡ whereȧ represents the derivative of a variable a with respect to time t, x(t) and v(t) are the two-dimensional position and velocity vector of the particle, m is the particle mass, is an asymmetric friction coefficient tensor with the friction constant γ and the magnetic field magnitude B, and ξ T (t) is the thermal noise described by a Gaussian white noise. The fluctuation-dissipation relation imposes ξ T (t)ξ T T (t ′ ) = 2γT Iδ(t − t ′ ) with the identity matrix I, where the superscript T denotes the transpose. The Under usual experimental conditions, the friction coefficient is so large that the inertia effects can be ignored. The equation of motion in this overdamped (small-mass) limit was obtained by integrating out Eq. (3) (see Appendix A and [39]) aṡ which is clearly singular at u = 0. The symmetric part of the correlation matrix Z is with indicating that the particle in the overdamped limit experiences a heat reservoir with the effective temperature T B lower than the original reservoir temperature T . This is consistent with the diffusion coefficient decrease for a charged Brownian particle in the presence of magnetic field, reported in [41]. The antisymmetric part of the correlation matrix plays a crucial role for featuring nonequilibrium-ness in this subtle overdamped limit, which generates a rotational probability current (curl flux), verified recently by numerical simulations for various systems [42][43][44][45]. This can be more transparent by deriving the probability current directly in the Fokker-Planck description.
The systematic way to obtain a small-mass expansion of the underdamped Fokker-Planck (Kramer) equation is well established [39,40,46]. Its leading order gives the overdamped Fokker-Planck (FP) equation governing the time evolution of the position distribution function p η (x, t) as [39] with the probability current where the subscript η denotes the non-white η T noise.
Note that the probability current has an asymmetric diffusion matrix T G −1 , which is consistent with the singular correlation matrix in Eq. (6) (explicitly shown in [39]) and is also responsible for the curl flux mentioned above. Nevertheless, the antisymmetric part of the diffusion matrix does not contribute to the time evolution of p η (x, t), as it is sandwiched between the same gradient operators in the FP equation. Thus, the distribution p η (x, t) should be identical to that for the (naive) overdamped system with the symmetric diffusion matrix (T B /γ)I, which is the symmetric part of the original diffusion matrix T G −1 .
This naive overdamped Langevin equation can be simply obtained by setting m = 0 in the underdamped equation of Eq. (3) aṡ with a white Gaussian noise ξ . The corresponding naive probability current is We will refer this naive dynamics as ξ-dynamics, whereas the original overdamped dynamics with the non-white noise as η-dynamics.
The equivalence of the distribution functions for both dynamics as p η (x, t) = p ξ (x, t) guarantees that the average of any position-dependent observable (internal energy, system Shannon entropy, etc) does not discriminate the true overdamped η-dynamics and the naive (incorrect) ξ-dynamics. However, the average of pathdependent observables such as currents could depend on the antisymmetric part of the diffusion matrix. One specific example has been already reported in Ref. [39] by an explicit calculation of the average work currents for two different dynamics for an exactly solvable model, which turn out to be clearly different from each other. Therefore, one should be careful in discussing the TUR involving the current average as well as its variance in the presence of a magnetic Lorentz force.

III. EXTENDED FP EQUATION AND ALTERNATIVE DYNAMICAL OBSERVABLE
Consider a general accumulated current Φ in the overdamped regime as where Γ = {x(t ′ )|t ′ ∈ (0, t)} denotes a trajectory in the state space, Λ(x) is an arbitrary state-dependent vector ('weight' function), and • represents the Stratonovich product.
It is convenient to consider the joint distribution p(x, Φ, t) for the position and the current, from which the current average and its fluctuations can be easily calculated. As an additional stochastic variable, the current satisfiesΦ Together with Eq. (11), one can derive the extended FP equation for the naive ξ-dynamics as with the FP operator where· denotes quantities and operators in the extended phase space and∇ Φ = ∇ + ∂ Φ Λ(x) is a tilted gradient operator. For detailed derivation, see Appendix B. By integrating out over Φ in Eq. (15), one can recover the ordinary FP equation with the probability current given in Eq. (12). Now, let us derive the extended FP equation for the original η-dynamics. As the η T noise is non-white, it is not straightforward to derive the extended FP operator directly. Thus, we again go back to the underdamped version and take the small-mass expansion. After a lengthy but straightforward calculation (shown in Appendix C), we obtain witĥ where the scalar function with Note that the gradient operator in Eq. (19) works only inside the scalar function (not on the distribution function). This scalar function is originated from the antisymmetric part of the diffusion matrix and contributes to the time evolution of the joint distribution, in contrast to the ordinary distribution case. Thus,p η (x, Φ, t) =p ξ (x, Φ, t) and thus A(Φ) η = A(Φ) ξ for an arbitrary function A(Φ).
Our key observation is that the extended FP opera-torL η,Φ has the same mathematical structure with the time evolution operatorL ξ,Ψ for the joint distribution p ξ (x, Ψ, t) with the alternative dynamic observable in the ξ-dynamics. In short, we find the operator correspondence relationL which is easily verified from Eq. (B5) in Appendix B for the explicit derivation ofL ξ,Ψ and Eq. (18). Note that the alternative dynamic observable Ψ and the original current Φ differ by the accumulated state-dependent observable ϕ, meaning that Ψ is not antisymmetric under time reversal in contrast to Φ. The correspondence relation between the extended FP operators implies that the random variables Φ in the η-dynamics and Ψ in the ξ-dynamics should be equivalent in their distributions at any time during the evolution, starting with the same initial distribution. In other words, we can identify the equivalence relations aŝ Using these relations, we can avoid complex calculations for the averages, caused by the singular nature of the nonwhite noise in the original η-dynamics. Figure 2 shows the schematic description of this relation. For example, the current average and its variance in the η-dynamics are identical to those of the alternative dynamic observable in the ξ-dynamics as which are useful to calculate the TUR factor Q Φ for the original η-dynamics.
The time evolution of the alternative dynamic observable Ψ reads from Eq. (20) yielding the relation between the average current rates It is worth mentioning that the non-zero average scalar term ϕ ξ represents the difference of two different averages of the current Φ. As in the traditional stochastic thermodynamics like in the ξ-dynamics, the average current rate in the ηdynamics is also written in an integral form involving the probability current (shown in Appendix D) as The average rate of Ψ in the ξ-dynamics is also written in a similar form, using Eq. (24), as with the effective probability current where we used Eq. (19) and the relation dx with the distribution equivalence of p η (x, t) = p ξ (x, t). This relation along with Eqs. (26) and (28) confirms Eq. (25).
To discuss the TUR, we consider the steady-state entropy production rate for the η-dynamics. Its stochastic version in the steady state can be written aṡ where the superscript 'ss' denotes a steady-state quantity such as J ss with the steady-state distribution p ss η (x) (see Appendix D.3 for its derivation). Using Eqs. (32) and (26), we obtain the average steady-state rate where the relation of Eq.
Using Eqs. (23) and (34), the TUR factor Q Φ for the η-dynamics is now expressed in terms of quantities in the ξ-dynamics as We refer to R Ψ as the alternative TUR factor for Ψ in the perspective of the ξ-dynamics. In the normal overdamped ξ-dynamics, the conventional TUR should be valid for general irreversible currents of the generic type in Eq. (13) with an arbitrary duration time t [9,10]. However, our alternative dynamic observable Ψ is not of the generic type, but is given by the combination of the current Φ and the accumulated state-dependent variable ϕ. Furthermore, ∆Σ is not the entropy production for the ξ-dynamics (see Eq. (D19) for the true entropy production). Thus, the alternative TUR factor R Ψ may violate the conventional TUR bound of 2k B . In the next section, we show that this violation indeed occurs and derive a new bound analytically.

IV. MODIFIED TUR
The derivation of the conventional TUR is based on the observation that the TUR is a special case of the Cramer-Rao (CR) inequality [47,48]. Here, we take a similar but slightly different approach from the conventional one adopted in Refs. [9,10], to derive a new lower bound for the alternative TUR factor R Ψ in the ξ-dynamics.
The CR inequality generally provides a lower bound of the variance of any dynamic observable O(Γ) in a stochastic process with a parameter θ as where O θ = dΓO(Γ)P θ (Γ) with the trajectory probability P θ (Γ) and the Fisher information is defined as If the dynamics of our interest is defined at θ = 0, it is crucial to find an appropriate modified dynamics with a nonzero θ, which yields lim θ→0 ∂ θ O θ = O 0 . Then, the (alternative) TUR factor in Eq. (35) naturally comes into the CR inequality at θ = 0. We take the ξ-dynamics as the unperturbed one at θ = 0. Similar to the perturbation term in the previous studies [9,10], we consider a linear perturbation on the force as The steady-state distribution p ss ξ (x) satisfies ∇ T · J ss ξ (x) = 0 and we also find ∇ T · J ss ξ (x) = 0 as well from Eq. (29). In the previous studies for the conventional TUR [9,10], γJ ss ξ (x) was used instead of GJ ss ξ (x) in the perturbation term of Eq. (38).
It is easy to see that the perturbed steady-state distribution is identical to the unperturbed one, i.e. p ss ξ,θ (x) = p ss ξ (x), as the perturbed steady-state probability current satisfies the steady-state condition of ∇ T · J ss ξ,θ (x) = 0. The perturbed effective probability current in the steady state becomes which yields Ψ ss ξ,θ = (1 + θ) Ψ ss ξ from Eq. (28), thus we find The remaining task is to express the Fisher information I(θ) at θ = 0 in terms of ∆Σ ss ξ . Using the Onsager-Machlup theory [49] in Eq. (11), the trajectory probability is written as where the initial probability p ss ξ,θ (x(0)) and the normalization factor N θ are θ-independent. Then, one can easily find the explicit form of the Fisher information from Eq. (37) as which is θ-independent. Using Eq. (34), we find Using Eqs. (35), (36), (41) and (44) along with Eq. (8) and setting θ = 0, we finally obtain the modified TUR for the overdamped η-dynamics as which is the main result of our paper. Note that we restored the Boltzmann constant k B here. The result shows that the Lorentz magnetic force always lowers the threshold of the TUR factor, which weakens the trade-off constraint. Obviously, the conventional TUR is recovered at B = 0. Interestingly, Chun et al [34] found the same form of the TUR lower bound in an exactly solvable linear model for the work and heat current in the long-time (t → ∞) limit. Our result applies to a general current in a general nonlinear system for an arbitrary duration time. In the next section, we derive the TUR factors exactly for the linear model for a finite duration time and confirm the validity of our modified TUR. The equality condition of the modified TUR can be determined by the equality condition of the Cramer-Rao inequality in Eq. (36) as After a straightforward calculation following a similar procedure in Ref. [9], we find that the equality holds with the two constraints as with an arbitrary constant µ, and Note that the entropy production current automatically satisfies the first constraint for B = 0, but not for B = 0 (see Eq. (32)). Other currents may satisfy the first constraint for some specific dynamics (see the example in the next section), but not in general. The second constraint is independent of the choice of currents, Λ(x), and more interesting. For B = 0, the second term in the right hand side of Eq. (48) vanishes, thus it reduces to the equality condition for the conventional TUR [9], which requires uniform local entropy production. This second constraint becomes trivially satisfied in the equilibrium limit where J ss η (x) vanishes (detailed balance). For B = 0, however, the second term exists and in fact dominates near equilibrium, as it is of the first order in J ss η (x), while the others, including Ṡ tot ss η , are of the second order. Hence, the equality condition is violated even in the equilibrium limit. Instead, the second constraint can be satisfied by tuning B (thus, G −1 a ) to make the second term also be of the second order in J ss η (x). In the next section, we will show this example explicitly, where the equality of the modified TUR holds out of equilibrium.
It is straightforward to extend our modified TUR to the η-dynamics with an arbitrary initial state. This generalization in the normal overdamped ξ-dynamics was reported recently in [50,51]. We also study this generalization in the η-dynamics and find the modified TUR with an arbitrary initial state for a finite duration time t as whereΦ(t) is the current at the final time t of duration and Eq. (45) is recovered in the steady state. The derivation details are given in Appendix E. A generalization to higher dimensional cases is also possible. The overdamped motion in the perpendicular plane to the magnetic field follows the same behavior discussed before with the effective temperature T B . In the parallel plane, the particle does not feel the Lorentz force and the normal overdamped motion is expected with the temperature T . The contribution to the Fisher information in Eq. (43) from this normal overdamped motion in the parallel plane is smaller as T > T B , thus the total Fisher information is smaller than the entropy production of the η-dynamics in Eq. (33). Therefore, our main result still holds in higher dimensions, even though the TUR equality condition can not be satisfied for non-zero B. For details, see Appendix F.

V. EXAMPLE
To confirm our main result, we consider an analytically solvable system with a linear force as The diagonal elements represent a force applied by a harmonic potential with spring constant k, while the offdiagonal elements indicate a nonequilibrium driving force with strength ǫ. This system has been investigated in various studies [34,37,39,[52][53][54], because it is analytically solvable. In the remaining text, we will omit the subscript η for brevity. We consider two accumulated currents of work W done on the system and heat Q dissipated into the reservoir, which are defined in Eq. (13) with weight finctions Since the model basically belongs to the Ornstein-Uhlenbeck process [40,55,56], the steady-state distribution p ss (x) should be Gaussian as with the covariance matrix C = xx T . From the steadystate condition, we can easily find in Eq. (G3) as which is stable for γk + ǫB > 0. The steady-state probability current is then obtained from Eq. (10) as The steady-state averages of work W and heat Q are calculated by Eq. (26) as which agrees with the result in [39]. The scalar function in Eq. (19) becomes a constant as ϕ(x) = 2ǫBT /(γ 2 +B 2 ) for both currents, which represents the difference between the true and the naive average in Eq. (25). One can also consider the total entropy production ∆S tot with Λ S (x) = GJ ss (x)/[T p ss (x)] in the steady state (see Eq. (32)). Using Eq. (55), we find the weight function as from which we obtain ∆S tot ss = Q ss /T and the corresponding scalar function ϕ S (x) = 2ǫB/(γ 2 + B 2 ), as expected.
The most complex task is to find the variance of work, heat, and enropy production. From the extended FP equation ofp(x, Φ, t) in Eqs. (17) and (18), we derive the formal solution for the variance in terms of auxiliary matrices (see Appendix G). Finding the explicit forms of the auxiliary matrices, we obtain the variance and the TUR factor in the steady state with a finite duration time t.
First, the TUR factor for work is given as where the dimensionless parameters are with τ = γ/k and the stability condition becomes 1 + ǫ 0 B 0 > 0. It monotonically increases with t 0 with the minimum, Q m W = 2/(1 + B 2 0 ) at t 0 = 0 + , which is exactly the same as the TUR lower bound in Eq. (45). This confirms the validity of our modified TUR.
Eq. (55) indicates that the work current obeys the first equality constraint in Eq. (47). In addition, it is easy to show that the second constraint in Eq. (48) is satisfied at ǫ 0 = B 0 , where the TUR equality should hold. From Eq. (59) with this condition, the TUR factor takes its minimum Q W = Q m W , which is independent of t 0 . Thus, the TUR lower bound is achieved out of equilibrium with ǫ 0 = 0. This may be useful in inferring the entropy production by measuring current statistics [22][23][24][25], which should be more feasible in nonequilibrium.
We check our analytic results numerically by simulating the underdamped dynamics with a very small mass m. Using a generalized velocity-Verlet algorithm [57], we perform numerical integrations of the underdamped Langevin equation in Eq. (3) with dimensionless mass m 0 = km/γ 2 = 0.001. Numerical data are plotted in Fig. 3 (a) for several values of ǫ 0 with B 0 = 0.5 and k = γ = T = 1. We used 10 7 samples to evaluate the average and the variance of work as well as the average entropy production. The numerical data are in an excellent agreement with Eq. (59) with a small deviation due to finite-mass corrections. Note that Q W takes the modified TUR bound value, independent of t 0 at ǫ 0 = B 0 = 0.5, as expected.
We also find the TUR factor for heat as which monotonically increases with t 0 for ǫ 2 0 −1−2ǫ 0 B 0 > 0 and decreases for ǫ 2 0 − 1 − 2ǫ 0 B 0 < 0. Interestingly, the heat TUR factor Q Q is always larger than the work TUR factor Q W as with the equality in the t 0 → ∞ limit. Thus, Q Q also satisfies the modified TUR. The equivalence of Q W = Q Q in the t 0 → ∞ limit was also reported in [34]. Numerical data for Q Q also agree with the analytic results very well, shown in Fig. 3 (b). The TUR equality can not be achieved for heat current, as the first equality constraint in Eq. (47) is not satisfied. The TUR factor for the entropy production (also called the entropy Fano factor [15]) is (63) which is always larger than the work TUR factor as Thus, Q S also satisfies the modified TUR. Interestingly, all three TUR factors take the same value of 2(1 + ǫ 2 0 )/(1 + ǫ 0 B 0 ) 2 in the limit of t 0 → ∞. The first equality constraint for Q S is satisfied only for B 0 = 0, where the friction tensor G becomes proportional to I. As the second equality contraint of Eq. (48) is satisfied at ǫ 0 = B 0 , we find the TUR equality only at B 0 = ǫ 0 = 0 with Q S = 2.
It is also interesting to see that lim t0→0 + Q S = 2, independent of the values of B 0 and ǫ 0 . This may hint a possibility of the short-time TUR for inferring the entropy production [23] even with the presence of a magnetic field. However, this does not come from the TUR equality constraint and may not hold with a nonlinear force. Figure 3 (c) shows this property of Q S and the consistency between the numerical data and the analytic results. Now, we make some remarks on near equilibrium with small ǫ 0 , where the TUR factors for work, heat, and the entropy production become Even in the equilibrium limit, the work current always violates the conventional TUR for finite t 0 , which implies that any simulation result should show a violation even with very small ǫ 0 and very long t 0 . In contrast, the heat current always satisfies the conventional TUR for small ǫ 0 , but its finite-ǫ 0 correction is huge, implying that it is almost impossible to reach the conventional TUR lower bound in any practical simulation near equilibrium. In the case of the entropy production, the TUR factor approaches the conventional bound from below for B 0 ǫ 0 > 0 and from above for B 0 ǫ 0 < 0. In the limit of t 0 → ∞ and ǫ 0 → 0, the TUR factor for all currents approaches 2 (the conventional TUR bound), independent of B 0 . Finally, we point out that a variant of the TUR including the dynamic activity for the underdamped dynamics [36,37] is not useful in the overdamped limit, as the dynamic activity diverges with m → 0 (see Eq. (40) in Ref. [37] for this specific example). We also note that the result from the linear response theory is not informative [32] as the Onsager matrix is symmetric in this example [56].

VI. CONCLUSION
We present the modified TUR for the Langevin systems subject to a static magnetic field in the overdamped limit. The system is described by the overdamped Langevin equation with a non-white singularly correlated noise. By using the extended Fokker-Planck equation with a current variable in addition, we find the alternative dynamics with the conventional white noise and an alternative dynamic observable, which generates the extended distribution function equivalent to the original one.
Utilizing the Cramer-Rao inequality, we derive the modified TUR, which turns out to be surprisingly simple with one scale factor on the conventional TUR bound. This factor is given by the ratio of the reduced effective temperature of the overdamped motion to the reservoir temperature. This TUR is universal in the sense that the lower bound is independent of system parameters. We emphasize that this TUR is valid for a finite duration time and for general currents. Our modified TUR shows that the magnetic field lowers down the bound and the standard TUR is recovered without the magnetic field. From the exactly solvable models, we confirm the validity of our results. We also find that the TUR lower bound can be reached out of equilibrium with the magnetic field, which may be useful in inferring the entropy production by measuring non-entropic current statistics in nonequilibrium. We also generalize our TUR for an arbitrary initial state.
Our analysis is limited to systems with isotropic thermal reservoirs. It could be interesting to extend it to systems with anisotropic baths such as microscopic heat engines with broken time-reversal symmetry.
We start from the formal solution of the underdamped equation, Eq. (3), as Then, the noise-noise correlation matrix Z m (t − t ′ ) = η m (t)η T m (t ′ ) can be easily obtained as for u > 0. Its Laplace transform becomes which becomes T G −1 in the m → 0 limit. Its inverse Laplace transform gives Z(u) = 2T G −1 δ(u) and Z(−u) = Z T (u) from Eq. (A3).

Appendix B: The extended FP equation for the overdamped ξ-dynamics
We consider the ξ-dynamics in Eq. (11) and the current dynamic equation of Eq. (14). The extended equation of motion can be rewritten aṡ whereẋ in Eq. (14) is replaced in terms of x and ξ TB by Eq. (11) and ξ TB (t)ξ T TB (t ′ ) = (2T B /γ)Iδ(t − t ′ ). The corresponding FP operator for the Langevin equation of Eq. (B1) with a Stratonovich-type multiplicative noise is well established, e.g. see the textbook [55], which yieldŝ From Eq. (B2), we finally obtain is the tilted gradient operator. We can derive the extended FP equation with the alternative dynamic observable Ψ given in Eqs. (20) and (24) for the ξ-dynamics. This new variable changes only the second component of A q (q) by adding ϕ(x). Therefore, we find the extended FP operator for this case aŝ We start with the extended FP equation for the underdamped dynamics and then take the overdamped (smallmass) limit to find the extended FP equation for the original η-dynamics. This procedure is quite similar to that for deriving the ordinary overdamped FP equation of Eqs. (9) and (10).
The underdamped equation of motion is given by Eq. (3) and the accumulated current is written as with a trajectory Γ = {x(t), v(t)|t ∈ (0, τ )}, of which the dynamic evolution is given bẏ As v is a state variable in the underdamped description, the extended equation of motion including Φ does not induce any multiplicative noise, in contrast to the overdamped ξ-dynamics. Thus, the type of stochastic calculus such as Ito and Stratonovich is meaningless for the FP operator.
Considering the underdamped equation of motion in Eq. (3) together with the above current dynamic equation, we can write the extended FP equation as where the extended FP operator is split into the reversible and the irreversible part aŝ , the velocity gradient operator ∇ v , and the magnetic-field matrix B = G a (the antisymmetric part of G). For convenience, we include the Lorentz term (Bv) in the reversible operator. Note that the information of the current variable Φ is fully contained in the tilted position gradient operator ∇ Φ in the exactly same way as in the overdamped case (Appendix B). Hence, we can easily expect that the overdamped (small-mass) limit of the extended FP equation should be the same as that for the ordinary FP equation of Eqs. (9) and (10), except for replacing ∇ by∇ Φ , which is presented in Eq. (18).
For completeness, we briefly sketch the standard procedure for the small-mass expansion [39,40,46]. The strategy is to find a series expansion of the distribution p(x, v, Φ, t) in terms of the orthonormal basis of the irreversible operator L irr . Then, the extended FP equation provides a hierarchy of coupled differential equations for the expansion coefficients known as the Brinkman's hirarchy [40,58]. In the small-mass limit, most of higherorder coefficients can be neglected in this expansion and the overdamped extended FP equation can be rather easily obtained.
First, it is convenient to rotate the operator L irr in a Hermitian form by introducinḡ which is the steady-state solution of L irr , satisfying L irr p ss (v) = 0.
In terms of the classical analogue of the bosonic ladder operators the rotated operator is written as which is reminiscent of the Hamiltonian of the quantum harmonic oscillator. Its orthonormal eigenfunctions ψ n1,n2 (v) are well known with two quantum numbers (n 1 , n 2 ), which are simply related to each other by the raising and lowering ladder operators, a ± . The groundstate eigenfunction is given asψ 0,0 (v) = p ss (v). It is useful to consider the same rotation on the reversible operatorL rev,Φ as The FP equation (C3) can be rewritten in terms of the rotated operators as We expandp(x, v, Φ, t) in terms of the orthonormal eigenfunctions as with the expansion coefficientĈ n1,n2 =Ĉ n1,n2 (x, Φ, t).
By putting this expansion form into Eq. (C11) and focussing on the lower-order terms for small m [39], we find The overdamped extended distributionp η (x, Φ, t) is obtained by integrating out the velocity variable of p(x, v, Φ, t). Using Eqs. (C12) and (C13) with the orthogonality property of the eigenfunctions, we find C 0,0 =p η (x, Φ, t). Combining Eqs. (C14) and (C15), we find the overdamped extended FP operator aŝ Appendix D: Stochastic thermodynamics for the η-dynamics In this appendix, we develop the stochastic thermodynamics for the η-dynamics. As the noise η T has a singular character in the noise-noise correlation, the framework of the stochastic thermodynamics should be reexamined. Thus, we start from the underdamped dynamics where the stochastic thermodynamics is well established.

FP operator and probability distribution
The underdamped FP equation with the Lorentz force is where the ordinary FP operators L rev and L irr are given in Eqs. (C4) and (C5) with the replacement∇ Φ by ∇.
With the same series expansion, we finally get the same equations as Eqs. (C14) and (C15) for the expansion coefficients C n1,n2 (x, t) = dΦĈ n1,n2 (x, Φ, t), again by re-placing∇ Φ with ∇. As we have and then the overdamped FP operator becomes The underdamped distribution for the small-mass limit is obtained from the expansion like in Eq. (C13) p(x, v, t) ≈ψ 0,0 (v) C 0,0ψ0,0 (v) + C T 4. entropy production for the ξ-dynamics It is worthwhile to derive explicitly the stochastic entropy production rate and its average in the naive ξdynamics. At the first glance, one may think that the result may be trivial because the ξ-dynamics is a standard overdamped dynamics with a white Gaussian noise. It turns out to be rather subtle due to the mismatch of the asymmetric friction tensor G and the symmetric diffusion matrix (T B /γ)I. When we take the distribution equivalence p ξ (t) = p ξ (x(t), t) = p η (x(t), t) into account, it is obvious that the time derivative of the stochastic system entropy has the same form as in Eq. (D16). Following the similar procedure as above, we geṫ where the heat dissipation rateQ(t) is defined in Eq. (D15) and Λ S (x, t) = GJ ξ (x, t)/(T p ξ (x, t)) with the effective probability current J ξ (x, t) defined in Eq. (30). From the relation between currents (J ξ (x, t) = J η (x, t)) in Eq. (31), we find that the stochastic entropy expressions for both dynamics are identical, i.e.Ṡ tot However, its average should be different as expected from Eq. (25). In the ξ-dynamics, we find where the scalar function ϕ S (x, t) = −T ∇ T G −1 a Λ S (x, t) with Λ S (x, t) = GJ ξ (x, t)/(T p ξ (x, t)) from Eq. (19) and the corresponding alternative dynamic observable Σ, defined asΣ(t) =Ṡ tot (t) + ϕ S (x(t), t), satisfies Σ ξ = Ṡ tot η , consistent with Eq. (25).
Thus, we get Using the expression for the trajectory probability in Eq. (42) with the same initial condition for any value of θ, we can also find the Fisher information from Eq. (37) as Setting θ = 0 and using Eq. (31) , we find where ∆S tot η is the total entropy production for duration time t and Eq. (D10) is used.
Finally, we have a modified TUR for the overdamped η-dynamics with an arbitrary initial state for a finite duration time t as where the Boltzmann constant k B is restored.
Consider a three-dimensional dynamics with a magnetic field in one direction. The two dimensional plane perpendicular to the magnetic field is spanned by the first and second component of the position vector x. Its third component describes the magnetic field direction. Then, the friction coefficient tensor in Eq. (4) is given by The derivation procedure of the modified TUR is basically the same as before in the two-dimensional case, so we list here only the differences. The symmetric part of the correlation matrix in Eq. (7) becomes and the naive probability current in Eq. (12) becomes For the ξ-dynamics, T B /γ should be replaced by D s everywhere.
The key replacement is in the trajectory probability in Eq. (42), which yields the Fisher information as where the third component of |J ss ξ (x)| 2 makes the inequality for T B < T . As the entropy production for the η-dynamics in Eq. (33) does not change in form, we finally get the same modified TUR in Eq. (45) for three dimensions. The extensions to higher dimensions and also for arbitrary initial states are obvious. But, due to the inequality in Eq. (F5), the TUR equality cannot be attained for nonzero B in three or higher dimensions.
with the drift matrix A = G −1 F and the asymmetric diffusion matrix D = T G −1 . We dropped the subscript η for brevity. It is easy to show that the covariant matrix C ≡ xx T = C T should satisfy in the steady state [40,55,56]. Then, we find where the dimensionless parameters are used with ǫ 0 = ǫ/k and B 0 = B/γ. We consider an accumulated current Φ of the generic type in Eq. (13) with Λ(x) = Jx. The linearity in x makes the exact calculation of their means and variances in the steady state rather simpler. The extended FP equation in Eqs. (17) and (18) is written as with the tilted gradient operator∇ Φ = ∇ + ∂ Φ (Jx). By multiplying a function of variables and integrating out over x and Φ on both sides of the extended FP equation, we can derive the dynamic equation of its average in terms of the averages of other functions. As we are interested in the steady-state average only, we take the initial conditionp(x, Φ, 0) = p ss (x)δ(Φ).
For the first moment of Φ, we can easily obtain, through simple integrations by parts, where Tr{X} stands for the trace of a matrix X. As φ is a constant of time, we get the average accumulated current as Φ = tφ. Of course, we can get the same result by integrating Eq. (D6). It is straightforward but rather involved to calculate Φ 2 . Here, we only sketch the calculation procedure. First, we obtain the dynamic equation for the second moment of Φ as ∂ t Φ 2 = 2 Tr{φD T Jt − J T AM(t) + J T DJC} (G6) with M(t) = Φxx T . This equation contains higher moments M(t) and its dynamic equation will again contain higher moments, but only 4th moments of the position variable x. Since p ss (x) is Gaussian as in Eq. (53), the standard Wick's theorem splits a 4th moment into a combination of second moments, which can be given by the covariant matrix C. Finally, the dynamic equation for M(t) is given by the following closed equation, This is a linear differential equation for M(t), so that the formal solution is given by where an auxiliary symmetric matrix H is defined by the relation ACJC + CJ T CA T = AH + HA T , and a matrix E(t) is defined by E(t) = e −At = e − γk+ǫB B 2 +γ 2 t cos(ωt) − sin(ωt) sin(ωt) cos(ωt) (G10) with ω = (kB − γǫ)/(B 2 + γ 2 ). By inserting the solution into Eq. (G6) and using the relation ∂ t Var[Φ] = ∂ t Φ 2 − 2φ 2 t, we obtain where the diffusion coefficient D Φ is given by By introducing another auxiliary symmetric matrix K, for convenience, such that we finally reach the formal solution The formal solution allows us to obtain the variance by solving the linear equations in the elements of H and K provided by Eq. (G9) and (G13). By determining the auxiliary matrices for J = W, Q, and S, we obtain the variance for work, heat, and entropy production.