Low-dissipation engines: Microscopic construction via shortcuts to adiabaticity and isothermality, the optimal relation between power and efficiency.

We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical formula of the efficiency at maximum power with explicit expressions of dissipation coefficients under the optimized protocols. When the relative temperature difference between the two baths in the cycle is insignificant, this expression satisfies the universal law of efficiency at maximum power up to the quadratic term of the Carnot efficiency. For large relative temperature differences, the efficiency at maximum power tends to be 1/2. Furthermore, we analyze the issue of power at any given efficiency for general low-dissipation engines and then obtain the supremum of the power in three limiting cases, respectively. These expressions of maximum power at given efficiency provide the optimal relations between power and efficiency which are tighter than the results in previous references.

Equilibrium free-energy differences from a linear nonequilibrium equality.

Extracting equilibrium information from nonequilibrium measurements is a challenge task of great importance in understanding the thermodynamic properties of physical, chemical, and biological systems. The discovery of the Jarzynski equality illumines the way to estimate the equilibrium free-energy difference from the work performed in nonequilibrium driving processes. However, the nonlinear (exponential) relation causes the poor convergence of the Jarzynski equality. Here, we propose a concise method to estimate the free-energy difference through a linear nonequilibrium equality which inherently converges faster than nonlinear nonequilibrium equalities. This linear nonequilibrium equality relies on an accelerated isothermal process which is realized by using a unified variational approach, named variational shortcuts to isothermality. We apply our method to an underdamped Brownian particle moving in a double-well potential. The simulations confirm that the method can be used to accurately estimate the free-energy difference with high efficiency. Especially during fast driving processes with high dissipation, the method can improve the accuracy by more than an order of magnitude compared with the estimator based on the nonlinear nonequilibrium equality.

Stochastic thermodynamics with odd controlling parameters.

Stochastic thermodynamics extends the notions and relations of classical thermodynamics to small systems that experience strong fluctuations. The definitions of work and heat and the microscopically reversible condition are two key concepts in the current framework of stochastic thermodynamics. Herein, we apply stochastic thermodynamics to small systems with odd controlling parameters and find that the definition of heat and the microscopically reversible condition are incompatible. Such a contradiction also leads to a revision to the fluctuation theorems and nonequilibrium work relations. By introducing adjoint dynamics, we find that the total entropy production can be separated into three parts, with two of them satisfying the integral fluctuation theorem. Revising the definitions of work and heat and the microscopically reversible condition allows us to derive two sets of modified nonequilibrium work relations, including the Jarzynski equality, the detailed Crooks work relation, and the integral Crooks work relation. We consider the strategy of shortcuts to isothermality as an example and give a more sophisticated explanation for the Jarzynski-like equality derived from shortcuts to isothermality.

General neck condition for the limit shape of budding vesicles.

The shape equation and linking conditions for a vesicle with two phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. 70, 2964 (1993)PRLTAO0031-900710.1103/PhysRevLett.70.2964; Phys. Rev. E 53, 2670 (1996)1063-651X10.1103/PhysRevE.53.2670], and then we use the shape equation and linking conditions to prove that this conjecture holds not only for axisymmetric budding vesicles, but also for asymmetric ones. Our study reveals that the mean curvature at any point on the membrane segments adjacent to the neck satisfies the general neck condition for the limit shape of a budding vesicle when the length scale of the membrane segments is much larger than the characteristic size of the neck but still much smaller than the characteristic size of the vesicle.

Shortcuts to isothermality and nonequilibrium work relations.

In conventional thermodynamics, it is widely acknowledged that the realization of an isothermal process for a system requires a quasistatic controlling protocol. Here we propose and design a strategy to realize a finite-rate isothermal transition from an equilibrium state to another one at the same temperature, which is named the "shortcut to isothermality." By using shortcuts to isothermality, we derive three nonequilibrium work relations, including an identity between the free-energy difference and the mean work due to the potential of the original system, a Jarzynski-like equality, and the inverse relationship between the dissipated work and the total driving time. We numerically test these three relations by considering the motion of a Brownian particle trapped in a harmonic potential and dragged by a time-dependent force.

Constitutive relation for nonlinear response and universality of efficiency at maximum power for tight-coupling heat engines.

We present a unified perspective on nonequilibrium heat engines by generalizing nonlinear irreversible thermodynamics. For tight-coupling heat engines, a generic constitutive relation for nonlinear response accurate up to the quadratic order is derived from the stalling condition and the symmetry argument. By applying this generic nonlinear constitutive relation to finite-time thermodynamics, we obtain the necessary and sufficient condition for the universality of efficiency at maximum power, which states that a tight-coupling heat engine takes the universal efficiency at maximum power up to the quadratic order if and only if either the engine symmetrically interacts with two heat reservoirs or the elementary thermal energy flowing through the engine matches the characteristic energy of the engine. Hence we solve the following paradox: On the one hand, the quadratic term in the universal efficiency at maximum power for tight-coupling heat engines turned out to be a consequence of symmetry [Esposito, Lindenberg, and Van den Broeck, Phys. Rev. Lett. 102, 130602 (2009); Sheng and Tu, Phys. Rev. E 89, 012129 (2014)]; On the other hand, typical heat engines such as the Curzon-Ahlborn endoreversible heat engine [Curzon and Ahlborn, Am. J. Phys. 43, 22 (1975)] and the Feynman ratchet [Tu, J. Phys. A 41, 312003 (2008)] recover the universal efficiency at maximum power regardless of any symmetry.

Stochastic heat engine with the consideration of inertial effects and shortcuts to adiabaticity.

When a Brownian particle in contact with a heat bath at a constant temperature is controlled by a time-dependent harmonic potential, its distribution function can be rigorously derived from the Kramers equation with the consideration of the inertial effect of the Brownian particle. Based on this rigorous solution and the concept of shortcuts to adiabaticity, we construct a stochastic heat engine by employing the time-dependent harmonic potential to manipulate the Brownian particle to complete a thermodynamic cycle. We find that the efficiency at maximum power of this stochastic heat engine is equal to 1-sqrt[T(c)/T(h)], where T(c) and T(h) are the temperatures of the cold bath and the hot one in the thermodynamic cycle, respectively.

Weighted reciprocal of temperature, weighted thermal flux, and their applications in finite-time thermodynamics.

The concepts of weighted reciprocal of temperature and weighted thermal flux are proposed for a heat engine operating between two heat baths and outputting mechanical work. With the aid of these two concepts, the generalized thermodynamic fluxes and forces can be expressed in a consistent way within the framework of irreversible thermodynamics. Then the efficiency at maximum power output for a heat engine, one of key topics in finite-time thermodynamics, is investigated on the basis of a generic model under the tight-coupling condition. The corresponding results have the same forms as those of low-dissipation heat engines [ M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck Phys. Rev. Lett. 105 150603 (2010)]. The mappings from two kinds of typical heat engines, such as the low-dissipation heat engine and the Feynman ratchet, into the present generic model are constructed. The universal efficiency at maximum power output up to the quadratic order is found to be valid for a heat engine coupled symmetrically and tightly with two baths. The concepts of weighted reciprocal of temperature and weighted thermal flux are also transplanted to the optimization of refrigerators.

Recent theoretical advances in elasticity of membranes following Helfrich's spontaneous curvature model.

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solutions to these equations and their corresponding configurations are demonstrated.

Antigenicity and functional properties of ß-lactoglobulin conjugated with fructo-oligosaccharides in relation to conformational changes.

Bovine ß-lactoglobulin (ß-LG) was conjugated with fructo-oligosaccharides (FOS) by Maillard reaction to investigate the relationship among antigenicity, functional properties, and conformational changes of ß-LG. When comparing the antigenicity of ß-LG conjugated with FOS at different ratios, the lowest antigenicity of ß-LG was observed at a ratio of 1:4, which was about 7 times lower than that of the control ß-LG. Thus, the ratio of 1:4 was chosen to conjugate ß-LG with FOS, and the functional properties and conformational changes of ß-LG-FOS conjugates were investigated. The functional properties (solubility, emulsifying ability, and emulsion stability) of ß-LG were enhanced after conjugation with FOS. Furthermore, the molecular weight of ß-LG increased from 18.4 to 19.9 kDa after conjugation with FOS, as evaluated by sodium dodecyl sulfate-PAGE and mass spectrometry. Partial unfolding of ß-LG occurred after conjugation with FOS, as reflected by the quenching of fluorescence, the red-shift of fluorescence spectra, and the increase of ß-strands, which may contribute to the decrease in antigenicity and the improvement of functional properties.

Low-dissipation heat devices: unified trade-off optimization and bounds.

We apply a unified and trade-off based optimization for low-dissipation models of cyclic heat devices which accounts for both useful energy and losses. The resulting performance regime lies between those of maximum first-law efficiency and maximum χ (a unified figure of merit corresponding to power output of heat engines). The bounds available for both symmetric and extremely asymmetric heat devices are explicitly obtained. The similarities for heat engines and refrigerators and the energetic advantages of the trade-off optimization are especially stressed.

Bounds and phase diagram of efficiency at maximum power for tight-coupling molecular motors.

The efficiency at maximum power (EMP) for tight-coupling molecular motors is investigated within the framework of irreversible thermodynamics. It is found that the EMP depends merely on the constitutive relation between the thermodynamic current and force. The motors are classified into four generic types (linear, superlinear, sublinear, and mixed types) according to the characteristics of the constitutive relation, and then the corresponding ranges of the EMP for these four types of molecular motors are obtained. The exact bounds of the EMP are derived and expressed as the explicit functions of the free energy released by the fuel in each motor step. A phase diagram is constructed which clearly shows how the region where the parameters (the load distribution factor and the free energy released by the fuel in each motor step) are located can determine whether the value of the EMP is larger or smaller than 1/2. This phase diagram reveals that motors using ATP as fuel under physiological conditions can work at maximum power with higher efficiency (> 1/2) for a small load distribution factor (< 0.1).

Coefficient of performance at maximum figure of merit and its bounds for low-dissipation Carnot-like refrigerators.

The figure of merit for refrigerators performing finite-time Carnot-like cycles between two reservoirs at temperature T(h) and T(c) (

Aggregation and conformational changes of bovine ß-lactoglobulin subjected to dynamic high-pressure microfluidization in relation to antigenicity.

Our previous research indicated that dynamic high-pressure microfluidization (DHPM) had a significant effect on the antigenicity of ß-lactoglobulin (ß-LG). In this study, aggregation and conformational changes subjected to DHPM (0.1-160 MPa) were investigated in relation to antigenicity. When DHPM pressure increased from 0.1 to 80 MPa, disaggregation of ß-LG samples and partial unfolding of the molecule were accompanied by an increase in ß-LG antigenicity, which was reflected in the decrease of particle size, increase of free sulfhydryl (SH) contents and ß-strands contents, and slight exposure of aromatic amino acid residues. At pressures above 80 MPa, the reaggregation of ß-LG may contribute to the decrease in antigenicity, which was reflected by an increase in particle size, the formation of aggregates, a decrease of in SH and ß-strands contents, and slight changes in aromatic amino acid residues. Aggregation and conformational changes of ß-LG under DHPM was related to its antigenicity.

Efficiency at maximum power output of linear irreversible Carnot-like heat engines.

The efficiency at maximum power output of linear irreversible Carnot-like heat engines is investigated based on the assumption that the rate of irreversible entropy production of the working substance in each "isothermal" process is a quadratic form of the heat exchange rate between the working substance and the reservoir. It is found that the maximum power output corresponds to minimizing the irreversible entropy production in two isothermal processes of the Carnot-like cycle, and that the efficiency at maximum power output has the form η(mP)=η(C)/(2-γη(C)), where η(C) is the Carnot efficiency, while γ depends on the heat transfer coefficients between the working substance and two reservoirs. The value of η(mP) is bounded between η(-)≡η(C)/2 and η(+)≡η(C)/(2-η(C)). These results are consistent with those obtained by Chen and Yan [J. Chem. Phys. 90, 3740 (1989)] based on the endoreversible assumption, those obtained by Esposito et al. [Phys. Rev. Lett. 105, 150603 (2010)] based on the low-dissipation assumption, and those obtained by Schmiedl and Seifert [Europhys. Lett. 81, 20003 (2008)] for stochastic heat engines which in fact also satisfy the low-dissipation assumption. Additionally, we find that the endoreversible assumption happens to hold for Carnot-like heat engines operating at the maximum power output based on our fundamental assumption, and that the Carnot-like heat engines that we focused on do not strictly satisfy the low-dissipation assumption, which implies that the low-dissipation assumption or our fundamental assumption is a sufficient but non-necessary condition for the validity of η(mP)=η(C)/(2-γη(C)) as well as the existence of two bounds, η(-)≡η(C)/2 and η(+)≡η(C)/(2-η(C)).

Compatibility between shape equation and boundary conditions of lipid membranes with free edges.

Only some special open surfaces satisfying the shape equation of lipid membranes can be compatible with the boundary conditions. As a result of this compatibility, the first integral of the shape equation should vanish for axisymmetric lipid membranes, from which two theorems of nonexistence are verified: (i) there is no axisymmetric open membrane being a part of torus satisfying the shape equation; (ii) there is no axisymmetric open membrane being a part of a biconcave discodal surface satisfying the shape equation. Additionally, the shape equation is reduced to a second-order differential equation while the boundary conditions are reduced to two equations due to this compatibility. Numerical solutions to the reduced shape equation and boundary conditions agree well with the experimental data [A. Saitoh et al., Proc. Natl. Acad. Sci. U.S.A. 95, 1026 (1998)].

Theoretical and numerical investigations on shapes of planar lipid monolayer domains.

Shapes of planar lipid monolayer domains at the air-water interface are theoretically and numerically investigated by minimizing the formation energy of the domains, which consist of the surface energy, line tension energy, and dipole electrostatic energy. The shape equation, which describes boundary curves of the domains at equilibrium state, is derived from the first order variation of the formation energy. A relaxation method is proposed to find the numerical solutions of the shape equation. The theoretical and numerical results are in good agreement with previous experimental observation. Some new shapes not observed in previous experiments are also obtained, which awaits experimental confirmation in the future.

Concise theory of chiral lipid membranes.

A theory of chiral lipid membranes is proposed on the basis of a concise free energy density which includes the contributions of the bending and the surface tension of membranes, as well as the chirality and orientational variation of tilting molecules. This theory is consistent with the previous experiments [J.M. Schnur, Science 264, 945 (1994); M.S. Spector, Langmuir 14, 3493 (1998); Y. Zhao,, Proc. Natl. Acad. Sci. USA 102, 7438 (2005)] on self-assembled chiral lipid membranes of DC8,9PC. A torus with the ratio between its two generated radii larger than sqrt[2] is predicted from the Euler-Lagrange equations. It is found that tubules with helically modulated tilting state are not admitted by the Euler-Lagrange equations and that they are less energetically favorable than helical ripples in tubules. The pitch angles of helical ripples are theoretically estimated to be about 0 degrees and 35 degrees, which are close to the most frequent values 5 degrees and 28 degrees observed in the experiment [N. Mahajan, Langmuir 22, 1973 (2006)]. Additionally, the present theory can explain twisted ribbons of achiral cationic amphiphiles interacting with chiral tartrate counterions. The ratio between the width and pitch of twisted ribbons is predicted to be proportional to the relative concentration difference of left- and right-handed enantiomers in the low relative concentration difference region, which is in good agreement with the experiment [R. Oda, Nature (London) 399, 566 (1999)].

Elasticity of polymer vesicles by osmotic pressure: an intermediate theory between fluid membranes and solid shells.

The entropy of a polymer confined in a curved surface and the elastic free energy of a membrane consisting of polymers are obtained by scaling analysis. It is found that the elastic free energy of the membrane has the form of the in-plane strain energy plus Helfrich's curvature energy [Z. Naturforsch. C 28, 693 (1973)]. The elastic constants in the free energy are obtained by discussing two simplified models: one is the polymer membrane without in-plane strains and asymmetry between its two sides, which is the counterpart of quantum mechanics in a curved surface [H. Jensen and H. Koppe, Ann. Phys. (N.Y) 63, 586 (1971)]; the other is the planar rubber membrane with homogeneous in-plane strains. The equations to describe equilibrium shape and in-plane strains of the polymer vesicles by osmotic pressure are derived by taking the first-order variation of the total free energy containing the elastic free energy, the surface tension energy, and the term induced by osmotic pressure. The critical pressure above which a spherical polymer vesicle will lose its stability is obtained by taking the second-order variation of the total free energy. It is found that the in-plane mode also plays an important role in the critical pressure because it couples with the out-of-plane mode. Theoretical results reveal that polymer vesicles possess mechanical properties intermediate between those of fluid membranes and solid shells.

Lipid membranes with free edges.

Lipid membrane with freely exposed edge is regarded as smooth surface with curved boundary. Exterior differential forms are introduced to describe the surface and the boundary curve. The total free energy is defined as the sum of Helfrich's free energy and the surface and line tension energy. The equilibrium equation and boundary conditions of the membrane are derived by taking the variation of the total free energy. These equations can also be applied to the membrane with several freely exposed edges. Analytical and numerical solutions to these equations are obtained under the axisymmetric condition. The numerical results can be used to explain recent experimental results obtained by Saitoh et al. [Proc. Natl. Acad. Sci. 95, 1026 (1998)].