This review focuses upon the measurement of force, indentation, and deformation with the atomic force microscope (AFM). Measurement and theory for elastic and viscoelastic particles and substrates are covered, as well as for deformable fluid drops and bubbles. A brief review is given of papers that use tapping mode imaging, normal and lateral force modulation, noise spectra, and indentation measurements. Measurement and calibration techniques that are essential for quantitative results with the AFM are discussed in detail. The author's contribution to elastic and viscoelastic theory for extended range forces is outlined, and the application of these to measured data for the adhesive van der Waals force and for the electric double layer repulsion is described. Contents

A Brownian particle subject to a time- and space-varying force is studied with the second entropy theory for nonequilibrium statistical mechanics. A fluctuation expression is obtained for the second entropy of the path, and this is maximized to obtain the most likely path of the particle. Two approaches are used, one based on the velocity correlation function and one based on the position correlation function. The approaches are a perturbation about the free particle result and are exact for weak external forces. They provide a particularly simple way of including memory effects in time-varying driven diffusion. The theories are tested against computer simulation data for a Brownian particle trapped in an oscillating parabolic well. They accurately predict the phase lag and amplitude as a function of drive frequency, and they account quantitatively for the memory effects that are important at high frequencies and that are missing in the simplest Langevin equation.

Several variational principles that have been proposed for nonequilibrium systems are analyzed. These include the principle of minimum rate of entropy production due to Prigogine [Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1967)], the principle of maximum rate of entropy production, which is common on the internet and in the natural sciences, two principles of minimum dissipation due to Onsager [Phys. Rev. 37, 405 (1931)] and to Onsager and Machlup [Phys. Rev. 91, 1505 (1953)], and the principle of maximum second entropy due to Attard [J. Chem.. Phys. 122, 154101 (2005); Phys. Chem. Chem. Phys. 8, 3585 (2006)]. The approaches of Onsager and Attard are argued to be the only viable theories. These two are related, although their physical interpretation and mathematical approximations differ. A numerical comparison with computer simulation results indicates that Attard's expression is the only accurate theory. The implications for the Langevin and other stochastic differential equations are discussed.

The author's nonequilibrium probability distribution is tested for time-varying mechanical work. Nonequilibrium Monte Carlo (NEMC) is used to simulate a Brownian particle in a soft-sphere solvent, driven by a moving external potential. Results are obtained for the phase lag and amplitude for drive frequencies ranging from the steady state to the transient regime. This now extends the application of the NEMC algorithm to a time-varying nonequilibrium system. The results are shown to agree with those obtained by nonequilibrium stochastic molecular dynamics and by Nosé-Hoover molecular dynamics, from which it is concluded that the nonequilibrium probability distribution correctly describes time-varying mechanical work and that it provides a fundamental basis for nonequilibrium statistical mechanics.

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