Adiabatic Processes Realized with a Trapped Brownian Particle

The ability to implement adiabatic processes in the mesoscale is of key importance in the study of artificial or biological micro- and nanoengines. Microadiabatic processes have been elusive to experimental implementation due to the difficulty in isolating Brownian particles from their fluctuating environment. Here we report on the experimental realization of a microscopic quasistatic adiabatic process employing a trapped Brownian particle. We circumvent the complete isolation of the Brownian particle by designing a protocol where both characteristic volume and temperature of the system are changed in such a way that the entropy of the system is conserved along the process. We compare the protocols that follow from either the overdamped or underdamped descriptions, demonstrating that the latter is mandatory in order to obtain a vanishing average heat flux to the particle. We provide analytical expressions for the distributions of the fluctuating heat and entropy and verify them experimentally. Our protocols could serve to implement the first microscopic engine that is able to attain the fundamental limit for the efficiency set by Carnot.

Figure 1

Illustration of the pseudoadiabatic and adiabatic processes. A Brownian particle of mass $m$ in a thermal bath at temperature $T$ moves in one dimension $x$ with velocity $v$ and is trapped with a harmonic potential $U(x)=\frac{1}{2}\kappa x^2$. The blue solid circle ${\mathrm{\Gamma }}_1$ represents the ensemble of microstates described by a Hamiltonian $\mathcal{H}(x,v)=\frac{1}{2}\kappa x^2+\frac{1}{2}m{v}^2$ with a given energy $\mathcal{H}(x,v)=E=kT$. Position and velocity are normalized by their standard deviation from the equipartition theorem, ${\sigma }_x=\sqrt{kT/\kappa }$ and ${\sigma }_v=\sqrt{kT/m}$. The red dashed ellipse ${\mathrm{\Gamma }}_2$ is the microstate set at the same energy but after two different adiabatic processes: (a) Pseudoadiabatic process, where $T_{\text{fin}}=2T$ and ${\kappa }_{\text{fin}}=2\kappa$ (i.e., $T/\kappa =\text{const}$); (b) adiabatic process, where $T_{\text{fin}}=2T$ and ${\kappa }_{\text{fin}}=4\kappa$ (i.e., $T^2/\kappa =\text{const}$). The arrows indicate the direction in which the process occurs.

Figure 2

Experimental setup and experimental protocols. (a) Sketch of the experimental setup. The kinetic temperature $T_{\mathrm{kin}}$ of a microparticle in an optical trap of stiffness $\kappa$ is controlled with a noisy electric field. (b) Pseudoadiabatic protocol. Kinetic temperature from the mean squared displacement, $\kappa ⟨x^2⟩/k$ (left axis, blue solid line), kinetic temperature from the calibration (left axis, blue dashed line) and stiffness of the trap (right axis, red dash-dot line) as functions of time. (c) The same as (b) but for the adiabatic process. Notice the larger fluctuations as $T_{\mathrm{kin}}$ increases.

Figure 3

Ensemble averages of the cumulative sums of thermodynamic quantities as a function of time in the pseudoadiabatic (a)–(b) and adiabatic (c)–(d) processes. (a) Energy as a function of time for the pseudoadiabatic process, $⟨W(t)⟩$ (blue), $⟨Q_x(t)⟩$ (red), $⟨\mathrm{\Delta }{E}_{\text{kin}}(t)⟩$ (green), $⟨Q(t)⟩$ (cyan), $⟨\mathrm{\Delta }U(t)⟩$ (magenta), and $⟨\mathrm{\Delta }E(t)⟩$ (black). (b) System entropy as a function of time for the pseudoadiabatic process, $⟨\mathrm{\Delta }{S}_x(t)⟩$ (blue) and $⟨\mathrm{\Delta }S(t)⟩$ (red). (c) Energetics of the adiabatic process. (d) System entropy change in the adiabatic process. Dashed curves are the theoretical values of the thermodynamic quantities obtained in the quasistatic limit.

Figure 4

Distribution of the heat absorbed by the particle in the position degree of freedom $Q_x$ for different thermodynamic processes: isothermal (blue circles), isochoric (red squares), pseudoadiabatic (green stars), and adiabatic (black crosses). The distributions are obtained from 900 cycles. The lines are theoretical distributions obtained from Eq. (3) using the initial and final values of kinetic temperature and stiffness used in the experiments, for the different processes: isothermal (solid line), isochoric (dashed line), pseudoadiabatic (dotted line), and adiabatic (dashed-dotted line).

Figure 5

Distribution of the system entropy change in the overdamped description, $\mathrm{\Delta }{S}_x$ (blue squares) and of the total system entropy change $\mathrm{\Delta }S$ (red circles) in the pseudoadiabatic (open symbols) and adiabatic (closed symbols) processes shown in Figs. 2 and 2. The distributions are obtained from 900 cycles. Theoretical distributions for $\mathrm{\Delta }{\mathcal{S}}_x$ [blue solid curve, Eq. (4)] and $\mathrm{\Delta }\mathcal{S}$ [red dashed curve, Eq. (5)] are also shown. All the quantities are shifted by their mean such that the mean of the represented quantities is zero.