Stochastic thermodynamics: work, heat and information for one small system

A bead in a harmonic trap. The bowl is the potential energy; the shaded curve is the Boltzmann distribution of positions, with width

\sqrt{k_BT/k}

. Original schematic, not measured data.

From one trapped bead to a fuel-driven enzyme

The first chapter measures how far a trapped bead wanders at room temperature. The second splits the energy of a single pull into work and heat. The third works out the average and the spread of that work when the trap moves at constant speed, and the fourth turns many irreversible pulls into an equilibrium free energy. The fifth measures the bead first and then moves the trap, so the information in each reading becomes work. The sixth turns to an enzyme that hops around a three-state cycle and finds the entropy it produces each second, and the seventh asks how much of that an observer who sees only part of the cycle can prove. The last chapter reads the 2003 paper that showed why the free energy from few pulls is biased.

You need algebra, the exponential function and a little probability. Each chapter introduces Gaussian distributions, averages and the Langevin equation when it first needs them. FR25

01 · L1
How far a trapped bead wanders at room temperature
Use the Boltzmann distribution to find how far a trapped bead wanders, how fast it forgets where it was, and why thermal energy sets the scale.
02 · L2
Work and heat for a single pull
Define work and heat along one noisy trajectory, check the first law for a single pull, and see why a sudden jump costs more than a slow drag.
03 · L2
The mean work and its spread when a trap moves at constant speed
Solve how far a bead lags behind a moving trap, derive the mean dissipated work and the work variance, and see how speed changes both.
04 · L3
How the Jarzynski equality and the Crooks relation recover free energy from irreversible pulls
Recover a free energy from the exponential average of work, find where forward and reverse work distributions cross, and see why a few pulls decide it.
05 · L3
How much work can be extracted by measuring a trapped bead and moving the trap
Derive the information in a noisy reading of a trapped bead, the mean work a trap shift extracts from it, and why a shift alone cannot use all of it.
06 · L3
How much entropy a fuel-driven enzyme produces each second
Write the master equation of a three-state enzyme cycle, show that entropy production is cycle rate times fuel, and see what counting cycles proves.
07 · L4
How much dissipation an observer can see when part of the motion is hidden
Bound entropy production from timed snapshots, show why a bound-or-empty record looks perfectly reversible, and meet the clues in waiting times.
08 · L4
Reading the 2003 paper by Gore, Ritort and Bustamante on the bias of free energies from few pulls
Read the three figures of the 2003 PNAS paper on finite-sample bias, reproduce its twenty-pull result and test it on simulated pulls.

Experiments along the way

01Drag a trapped beadPull a bead uphill many times and compare the average work, the Jarzynski estimate and the crossing of forward and reverse pulls.02Run an information engineMeasure a trapped bead, shift the trap toward the reading and compare the extracted work with the information gained.03Hide an enzyme’s dissipationWatch a fuel-driven enzyme cycle and compare its entropy production with what cycle counts, snapshots or a bound-or-empty signal can prove.04Calibrate an optical trapMeasure trap stiffness from the thermal motion of a free bead.

The dragged-trap experiment simulates a one-micrometre bead in water at 300 K with exact relaxation steps between small trap moves. Its work values are simulated; the exact free-energy change is shown beside them as a separate reference.

Sources and what we recomputed

U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, arXiv:1205.4176v1; Reports on Progress in Physics 75, 126001 (2012)
Version 1, 18 May 2012. Section 2: overdamped Langevin dynamics, Eq. (14) for the heat released into the medium and Eqs. (15)–(19) for the first law along one trajectory. Section 4: Eqs. (58) and (61), the Jarzynski and Crooks relations. Section 6.2: entropy production for master equations, Eqs. (120)–(129). Section 7.2: relative-entropy bounds on dissipation, Eqs. (146)–(150). Section 7.3: measurement and feedback, Eqs. (151)–(163).
O. Mazonka and C. Jarzynski, Exactly solvable model illustrating far-from-equilibrium predictions, arXiv:cond-mat/9912121v1 (1999)
Version 1, 7 December 1999. A bead dragged through a viscous fluid by a harmonic spring at constant speed. Eqs. (4)–(7) set up the model and the work rate; Eq. (26) defines the finite-time factor; Eqs. (43)–(53) add a constant force, give the free-energy change in Eq. (49) and the Gaussian mean and variance of the work in Eqs. (51)–(53). We recompute these results; the paper does not give our numerical examples.
C. Jarzynski, A nonequilibrium equality for free energy differences, arXiv:cond-mat/9610209v1; Physical Review Letters 78, 2690–2693 (1997)
Version 1, 30 October 1996. The equality between the exponential average of work over repeated switching from equilibrium and the equilibrium free-energy difference. We read the abstract and the setup; the derivation in this course is the Langevin version given in the Seifert review.
G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, arXiv:cond-mat/9901352v4; Physical Review E 60, 2721–2726 (1999)
Version 4. The ratio of forward and time-reversed work distributions. We use the relation in the form of Seifert Eq. (61) and test it on simulated pulls; we did not reproduce the paper’s own examples.
J. Gore, F. Ritort and C. Bustamante, Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements, PNAS 100, 12564–12569 (2003)
Open-access text on PubMed Central. Figure 1: Gaussian dissipated-work distributions with means of 0.5, 1, 2 and 4 thermal energy units. Figure 2: 100 000 estimates, each from 20 work values, at a mean dissipated work of 4 units; mean bias about 1.07 units. Figure 3: bias against sample size and the fall-off exponent for the constant C equal to 15. We reproduce the Figure 2 bias by our own sampling.
V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nature Physics 8, 143–146 (2012)
A single 2.94 μm colloidal bead in an optical trap whose stiffness cycles between 281 and 1180 fN/μm, with the bath switched between 22 °C and 86 °C by laser heating. In the 76 °C data the work of single cycles fluctuates strongly; the measured efficiency approaches 0.14 for slow cycles, about 0.91 of the Carnot value, and the efficiency at maximum power, about 0.075, is close to the Curzon–Ahlborn value 0.081. Cited as an experimental reading example only.
T. Sagawa and M. Ueda, Generalized Jarzynski equality under nonequilibrium feedback control, arXiv:0907.4914v3; Physical Review Letters 104, 090602 (2010)
Version 3, 20 January 2010. Eq. (3): with feedback, the mean work can fall below the free-energy change by at most the thermal energy times the mutual information. Eq. (4): the corresponding equality for a process that starts in equilibrium. The trapped-bead engine of this course and its numbers are our own.
T. Sagawa and M. Ueda, Minimal energy cost for thermodynamic information processing: measurement and information erasure, arXiv:0809.4098v3; Physical Review Letters 102, 250602 (2009)
Version 3, 30 May 2009. Eq. (4): the work to measure and then erase the record together is at least the thermal energy times the mutual information obtained.
A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, arXiv:1502.05944v2; Physical Review Letters 114, 158101 (2015)
Version 2, 30 April 2015. Eq. (4): for a biased random walk the dissipation times the squared relative uncertainty equals A coth(A/2), at least two thermal energy units. Eqs. (7) and (10): the diffusion constant and the product Q for a general network. Eq. (19): the three-state ATP hydrolysis cycle. Eq. (20): the sharper bound for a ring of N states. The enzyme rates and numbers in this course are our own.
I. A. Martínez, G. Bisker, J. M. Horowitz and J. M. R. Parrondo, Inferring broken detailed balance in the absence of observable currents, arXiv:1809.02084v2 (2019)
Version 2, 23 July 2019. Eq. (1): the entropy production rate is at least the relative entropy rate between an observed trajectory and its time reverse. Eq. (2): that rate splits into an affinity part and a waiting-time part (Eqs. 3 to 6). The partially hidden network example observes states 1 and 2 and one hidden class. The journal version was not checked.