Thermodynamic Bounds on Precision in Ballistic Multi-Terminal Transport

For classical ballistic transport in a multi-terminal geometry, we derive a universal trade-off relation between total dissipation and the precision, at which particles are extracted from individual reservoirs. Remarkably, this bound becomes significantly weaker in presence of a magnetic field breaking time-reversal symmetry. By working out an explicit model for chiral transport enforced by a strong magnetic field, we show that our bounds are tight. Beyond the classical regime, we find that, in quantum systems far from equilibrium, correlated exchange of particles makes it possible to exponentially reduce the thermodynamic cost of precision.

Heisenberg's uncertainty principle is a paradigm example for the ubiquitous interplay between fluctuations and precision. It entails that the accuracy of simultaneous measurements of non-commuting observables is subject to a fundamental lower bound arising from intrinsic fluctuations in the underlying quantum states. Quite remarkably, the precision of non-equilibrium thermodynamic processes might be restricted through thermal fluctuations in a similar way: Barato and Seifert recently suggested that steady-state biomolecular process are subject to a universal trade-off between entropy production and dispersion in the generated output [1]. Since its discovery, this thermodynamic uncertainty relation has triggered significant research efforts. A general proof based on methods from large-deviation theory was given by Gingrich et al. for Markov jump processes satisfying a local detailed balance condition [2,3]. Further developments include extensions to finite-time [4,5] and discrete-time [6] processes, Brownian clocks [7] and systems obeying Langevin dynamics [8,9].
In light of these results, the question arises, whether a fundamental bound on the precision of thermodynamic processes can be derived from first principles. An ideal stage to investigate this problem is provided by ballistic conductors, that is, devices, whose dimensions are smaller than the mean free path of transport carriers. In such systems, the transfer of particles is governed by reversible laws of motion, while all irreversible effects are relegated to external reservoirs, a mechanism also know as moderate damping [11,12]. This structural simplicity does not only enable the use of physically transparent models; it also leads to a direct link between micro-dynamics and thermodynamic observables. Features such as the inertia of carriers or Lorentz-type forces, which are not covered by Markov jump processes or overdamped diffusion, are thereby naturally included. These advantages have made ballistic models an important source of insights on classical [13][14][15][16] and quantum [17][18][19][20][21] transport mechanisms. Here, we use this framework to derive a thermodynamic uncertainty relation for classical ballistic transport, which can be traced back to elementary properties of Hamiltonian dynamics.
FIG. 1. Four-terminal setup as an example for multi-terminal ballistic transport. A two-dimensional target is connected to four reservoirs denoted by their chemical potentials µ1, . . . , µ4 via perfect leads of width l1, . . . , l4. The solid line crossing the conductor shows the trajectory of a classical particle with energy E, which enters the target region at the point ζ in E ≡ (τ in , p in τ ) in the reduced phase space and leaves it at ζ out E ≡ (τ out , p out τ ) after being deflected by the target potential and the magnetic field B. The coordinate τ parameterizes the boundary of the target region and pτ denotes the corresponding canonical momentum. Since the particle follows Hamiltonian laws of motion, the scattering map (2) is one-to-one. Because τ and pτ are canonical variables, the Poincaré-Cartan theorem implies that this map is also volume preserving [10].
Scattering theory provides a powerful tool to describe ballistic transport in both, the classical and the quantum regime. In this approach, the conductor is modeled as a target that is connected to N perfect, effectively infinite leads. Each lead is attached to a reservoir with fully transparent interface injecting thermalized, non-interacting particles. Once inside the conductor, the particles follow deterministic dynamics until they are absorbed again into one of the reservoirs, Fig. 1.
On the classical level, the current flowing in the lead α towards the target corresponds to a phase-space variable J α [ξ t ], where the vector ξ t contains the positions and momenta of all particles in the conductor at the time t. In the steady state, the mean value and fluctuations of this current are given by where the average ⟨•⟩ has to be taken over the ensemble of trajectories of injected particles [22]. Exploiting that the injected particles are statistically independent and non-interacting, the expressions (1) can be made more explicit. Focusing on two-dimensional systems from here onwards, to this end, we decompose the trajectory of a single particle with energy E into an in- which arises due to heat dissipation in the reservoirs. Thus, σ can be regarded as the thermodynamic cost of the transport process, which is driven by the dimensionless thermodynamic forces F α ≡ (µ α − µ) (k B T ) with µ denoting a reference chemical potential.
We will now show that this cost puts a universal lower bound on the relative uncertainty [1] of each individual current. To this end, we consider the quadratic form where x, ψ ∈ R. For systems without an external magnetic field, A α can be written as Here, we used that, at vanishing magnetic field, the transmission coefficients obey T αβ E = T βα E as a consequence of time-reversal symmetry [22,[25][26][27]. Next, we observe that, for any x, the second sum in (9) is non-negative if 0 ≤ ψ ≤ 2 [28]. Hence, under this condition, the quadratic form A α is positive semidefinite, since the first sum in (9) is generally non-negative. Consequently, setting ψ = 2 in (8) and taking the minimum with respect to x yields For systems, where time-reversal symmetry is broken by means of an external magnetic field, the transmission coefficients T αβ E,B are in general not symmetric with respect to α and β. However, they still fulfill the weaker constraint ∑ β T αβ E,B = ∑ β T βα E.B , which follows from the volume-preserving property of the scattering map (2) [13,22,24]. Using this sum rule, the quadratic form (8) can be expressed as where V αβ B ≥ 0 is defined analogous to V αβ in (9). Minimizing the term inside the curly brackets shows that the second sum is (11) is non-negative for any x if 0 ≤ ψ ≤ min y∈R (1 − e y + ye y )(e y + 1) (e y − 1) 2 ≡ ψ * ≃ 0.89612. (12) Moreover, the first contribution in (11), which does not depend on x, is non-negative due to the convexity of the Chiral transport. Bouncing orbits enforced by a strong magnetic field B sustain clockwise oriented currents between adjacent leads (inset). From top to bottom, the blue blue lines show the cost-precision ratio Q1 defined in (15) as a function of the rescaled bias increment F for systems with N = 2, . . . , 25 terminals and for the limiting case N → ∞. Interpolating between the minima of Q1, the dotted line crosses the two dashed lines, respectively indicating the bounds (10) and (13), at N = 2 and N → ∞. exponential function. Hence, by using the same argument as in the derivation of (10), we arrive at The bounds (10) and (13) constitute our first main result. Following from elementary microscopic principles, respectively, time-reversal symmetry and the conservation of phase-space volume, they hold for any scattering potential, any number of terminals and arbitrary far from equilibrium. On the macroscopic level, they imply that any increase in the precision 1 ε α , at which particles are extracted from the reservoir α, inevitably leads to a proportional increase of the minimal thermodynamic cost σ of the transport process. The symmetric bound (10) thereby has exactly the same form as the recently discovered thermodynamic uncertainty relation for Markov jump processes [1][2][3]. Remarkably, (13) shows that the minimal cost of precision is reduced by a factor ψ * 2 in systems, where an external magnetic field breaks the time-reversal symmetry of scattering paths.
To show that our bounds are tight, we consider an Nterminal conductor with flat target potential. An external magnetic field B forces incoming particles with mass m and charge q on bouncing orbits along the boundary of the target region, Fig. 2. This scattering mechanism is captured by the transmission coefficients (14) with periodic indices α, β = 1, . . . , N [13]. Here, the direction of the magnetic field has been chosen such that the Larmor circles with radius For a strong magnetic field, the typical Larmor radii are small compared to the dimensions of the conductor. Under this condition, the transmission coefficients are given by (14) throughout the relevant range of energies. Due to the asymmetric structure of these coefficients, a chiral steady state emerges, where currents flow in clockwise direction between neighboring reservoirs [21]. To generate a net transfer of particles, an external bias has to be applied breaking the N -fold rotational symmetry of the system. For simplicity, here we choose the chemical potentials of the reservoirs to increase linearly in steps proportional to 1 N , that is we set F α ≡ αF N . The mean currents and fluctuations can then be evaluated explicitly by inserting (14) into (4). Using the abbreviation E ≡ exp[F N ], we thus obtain the expressions for the dimensionless product Q α ≡ σε α k B of total dissipation (6) and relative uncertainty (7). In the simplest case N = 2, the transmission coefficients (14) are still symmetric and (15) reduces to Hence, Q α reaches its minimum at F = 0 and the bound (10) is saturated in the linear response regime. As N increases, the minimum of Q 1 becomes successively smaller and shifts to negative values of F, Fig. 2. For large N , we obtain the asymptotic expression lim N →∞ which should be compared with (12). In fact, (17) reaches its minimal value ψ * at F ≃ −1.49888. This result shows that the cost-precision ratio Q 1 can come arbitrary close to its lower bound (13) as the number of terminals increases. By contrast, Q α>1 asymptotically grows as N 2 at any F ≠ 0. This divergence is a consequence of the chiral transmission coefficients (14) enabling the exchange of particles only between clockwise-adjacent reservoirs: the currents J α>1 are effectively driven by the bias F N and hence vanish as 1 N , while the fluctuations S α>1 and the total dissipation σ stay finite for large N . So far, we have shown that precision in classical ballistic transport requires a minimal thermodynamic cost, which can be substantially reduced in systems with broken time-reversal symmetry. Although our derivations were performed in a 2-dimensional setting, it is straightforward to establish (10) and (13) also in 1 and 3 dimensions. Rather then spelling out the details of this procedure, in the last part of this article, we develop a perspective beyond the classical regime.
For a quantum theory of ballistic transport, the phasespace variable J α [ξ t ] in (1) has to be promoted to an operator in the Heisenberg picture. Replacing classical trajectories with quantum states, the ensemble average in (1) can then be evaluated using standard techniques from quantum scattering theory [29,30]. In this formalism, the crucial role of the scattering map (2) is played by the complex scattering matrices S αβ E,B , which connect the amplitudes of incoming waves in the lead β and outgoing waves in the lead α, respectively [31]. For fermionic particles, the mean current is thus obtained as Notably, this expression has the same structure as its classical correspondent (4) with the quantum transmission coefficients defined aŝ 19) and the Maxwell-Boltzmann distribution (5) replaced by the Fermi-Dirac distribution The anatomy of current fluctuations in the quantum regime is, however, more complicated than in the classical case; S α = S cl α − S qu α involves two components [31] both of which are non-negative. Depending only on single-particle quantities, S cl α can be regarded as the quantum analogue of the classical expression (4) with additional Pauli-blocking factors accounting for the exclusion principle. By contrast, the contribution S qu α , which is of second order in the transmission matrices T αβ E,B and hence describes the correlated exchange of two particles, has no classical counterpart [31].
The two-component structure (21) of the current fluctuations suggests to divide the relative uncertainty ε α = ε cl α − ε qu α into a quasi-classical part ε cl α ≡ S cl α J 2 α and a quantum correction ε qu α ≡ S qu α J 2 α . By following the lines leading to (10) and (13) it is then possible to establish the bounds [22,32] σε cl α ≥ 2k B and σε cl respectively for quantum systems with and without timereversal symmetry, where σ = k B ∑ α F α J α . As in the classical case, this result follows from the symmetrŷ T αβ E =T βα E of the quantum transmission coefficients (19) for B = 0 and from the sum rules ∑ βT αβ E,B = ∑ βT βα E,B for B ≠ 0 [31]. It implies in particular that the classical relations (10) and (13) are recovered close to equilibrium, i.e., for small affinities F α , and in the semi-classical regime, where the fugacities ϕ α ≡ exp[µ α (k B T )] are small [27]; in both cases the quantum fluctuations S qu α are negligible. In general, however, the quantum corrections ε qu α will spoil the bounds (10) and (13) as the following simple model shows. Consider a two-terminal conductor with narrow leads allowing only for a single open transport channel, i.e., the system is effectively 1-dimensional and the scattering matrices S αβ E,B reduce to complex numbers. The target acts as a perfect energy filter, which is fully transparent in a small window ∆ around the reference chemical potential µ and opaque at all other energies. Such filters are standard tools in mesocopic physics [33][34][35] and can be implemented, for example, with quantum Hall edge states [36]. Setting F 1 ≡ −F 2 ≡ F 2 and neglecting second-order corrections in ∆ (k B T ), we obtain by evaluating (18) and (21). Hence, while the product of total dissipation σ and quasi-classical uncertainty ε cl α is bounded by 2k B , the full cost-precision ratio σε α can become arbitrary small. Specifically, as F becomes large, the current J α saturates to a finite value, σ grows linearly and the fluctuations S α are exponentially suppressed. This example shows that a combination of quantum effects and energy filtering makes it possible to exponentially reduce the minimal thermodynamic cost of precision. Whether or not this phenomenon can be captured in a generalized trade-off relation, where either cost or precision enters non-linearly, remains an intriguing question for future research. Further prospects include the extension of our theory to systems with temperature gradients or bosonic particles.
Notably, the number ψ * , which enters the nonsymmetric bounds (13) and (22), also appears in a recently found trade-off relation between power and efficiency of stochastic heat engines [37]. These figures are indeed connected with the minimal cost of precision [38]. Using our approach, it might thus be possible to bound the performance of ballistic thermoelectric engines, a class of devices that is currently subject to active investigations, see for example [13, 14, 17-21, 33-36, 39]. At this point, we conclude by stressing that any violation of our classical bounds constitutes a clear signature of quantum effects. Therefore, our work provides a valuable new benchmark to probe non-classical transport mechanisms in future theoretical and experimental studies.

I. CLASSICAL BALLISTIC TRANSPORT
Starting from the definition Eq. 1, we derive the expressions Eq. 4 for the mean currents J α and the current fluctuations S α in sections A and B. In section C, we show that the transmission coefficients defined in Eq. 3 obey the relations as a consequence of time-reversal symmetry and conservation of phase-space volume, respectively.

A. Mean Currents
To implement the classical scattering formalism, we assume that, at t = 0, the target region is empty and each lead is filled with an ideal gas in equilibrium at the temperature and chemical potential of the attached reservoir. This configuration then evolves under Hamiltonian dynamics. Since the leads are considered infinite, the long-time limit can be taken without specifying the mechanism of particle exchange with the reservoirs [1]. Hence, the ensemble average in Eq. 1 becomes an average over the initial number of particles in each lead and the state ξ 0 of the system at the initial time t = 0 [1]. Introducing the phase-space variable n α [ξ t ], which counts the number of particles in the lead α, we thus have To rewrite this expression in terms of single-particle quantities, we first decompose the phase-space vector ξ t into single-particle components, i.e., Here, the vector ξ αj t contains the position and momentum coordinates of the particle j initially located in the lead α. Second, we note that with π α [ξ βj t ] = 1 if the particle indexed by β and j is located in the lead α at the time t and π α [ξ βj t ] = 0 otherwise. Third, since the particles follow non-interacting, deterministic dynamics, the single-particle sate ξ αj t at the time t is uniquely determined by the corresponding initial state, i.e., where M t,B is a one-to-one map in the single-particle phase space depending parametrically on the magnetic field B. Hence, using that the gas in each lead is initially described by a grand canonical ensemble, we obtain Here, H denotes the single-particle Hamiltonian and integrals with lower index extend over the region in phase space, where the particle is located in the respective lead. For the second expression in (2), we have used that all particles originating from the same lead give the same contribution to the mean current. Therefore, the index j on the single-particle phase-space vector can be dropped.
In a two-dimensional setup, the initial position in phase space of a particle approaching the target is determined by four parameters: its energy E ∈ [0, ∞), the time t in ∈ [0, ∞) the particle needs to reach the target and the coordinates ζ in 2mE], at which the particle enters the target region. Here, L denotes the circumference of the target region. For the definition of τ and p τ , see Fig. 1 of the main text. In this parameterization the mean current (2) becomes where K t,B ∶ (E, t in , ζ in E ) ↦ ξ α t denotes a one-to-one mapping from the initial condition specified by the parameters E, t in and ζ in E to the single-particle state ξ α t evolving from it. Note that this map is volume preserving for any t ≥ 0 by virtue of the Poincaré-Cartan theorem [2].
To carry out the long-time limit, we observe that, for large t, the typical time particles spend inside the target region becomes negligible. Therefore, we can replace K t,B with K ∞,B in (3). Invoking the relation with the scattering map S E,B defined in Eq. 2 and the definitions Eq. 3 and Eq. 5, we thus arrive at The second line thereby follows from the sum rule The fluctuations S α can be computed along the same lines as the mean currents. First, using the notation of the previous section, we have Second, evaluating the grand-canonical average yields Third, using that π 2 α = π α and repeating the steps leading from (2) to (4) gives where the second line follows from the sum rule

C. Transmission Coefficients
The symmetries (1) of the classical transmission coefficients result from the properties of the scattering map connecting the reduced phase-space coordinates, at which a particle with energy E enters and leaves the target region. First, we have with R ∶ ζ E ↦ R[ζ E ] denoting the time-reversal map, which switches the sign of momentum coordinates while leaving spatial coordinates unchanged. Thus, a mirror particle injected at τ out with energy E and transversal momentum −p out τ , under the reversed magnetic field −B, retraces the trajectory of the original particle and leaves the target region at τ in with transversal momentum −p in τ . Consequently, it follows that Here, we used in the first step that, for Hamiltonian dynamics, the scattering map S E,−B is invertible and volume-preserving due to the Poincaé-Cartan theorem [2]. Furthermore, we exploited that the domain of the reduced phase-space integrals is invariant under the timereversal map R. In the second step, we applied (5) and (6). Finally, in the third step, we relabeled the integration variables.
To derive the sum rules (5), we first note that where l α is the width of the lead α. Second, using again that the scattering map is invertible and volumepreserving, we have Comparing (7) and (8) yields the sum rules (1).

II. QUANTUM BALLISTIC TRANSPORT
We derive the bounds Eq. 22 on the quasi-classical part of the relative uncertainty, ε cl α ≡ S cl α J 2 α . To this end, we first note that, for B = 0, the quantum transmission coefficients Eq. 19 are symmetric, i.e.,T αβ E =T βα E [3]. Hence, using the expression Eq. 18 for the mean currents J α , the total rate of entropy production can be written as Together with Eq. (21), this structure implies where D αβ = F α − F β as in the main text. Furthermore, the coefficients W αβ ≡ ∫ non-negative, since Fermi functions f α E defined in Eq. 20 are bounded between 0 and 1. Expression (9) has the same structure as its classical counterpart Eq. 9. It follows that A cl α ≥ 0 for any x provided that 0 ≤ ψ ≤ 2. Thus, we end up with σε cl α ≥ 2k B . For B ≠ 0, the sum rules ∑ βT αβ E,B = ∑ βT βα E,B [3] can be used to express the total rate of entropy production as Here, the first integral, which does not depend on x, is non-negative, since the function g[y] is convex and ∂ y g[y] = f [y], for details see [4]. Therefore, the quadratic form (10) where w[y, z] ≡ (y +z) (y +1 z) ln (y +z) (1 y +z) −2 ln[y] z .