Temperature is a well-defined quantity for systems in equilibrium. For glassy systems, it has been extended to the non-equilibrium regime, showing up as an e ective quantity in a modified version of the fluctuation-dissipation theorem. However, experimental evidence supporting this definition remains scarce. Here, we present the first direct experimental demonstration of the e ective temperature by measuring correlations and responses in single molecules in non-equilibrium steady states generated under external random forces. We combine experiment, analytical theory and simulations for systems with di erent levels of complexity, ranging from a single bead in an optical trap to two-state and multiple-state DNA hairpins. From these data, we extract a unifying picture for the existence of an e ective temperature based on the relative order of various timescales characterizing intrinsic relaxation and external driving. Our study thus introduces driven small systems as a fertile ground to address fundamental concepts in statistical physics, condensed-matter physics and biophysics. F rom living cells to stars, natural processes occur under non-equilibrium conditions. Non-equilibrium systems have been classified, phenomenological patterns identified and theoretical frameworks developed, yet our current understanding of the fundamentals of the non-equilibrium problem still remains incomplete, undoubtedly far beyond what we know for equilibrium systems. Major advances in the field are the understanding of linear irreversible thermodynamics and, more recently, the development of fluctuation theorems 1-4 and macroscopic fluctuation theories 5,6 .Temperature, the measure of warmth or coldness of a system, is a genuine statistical concept. Related to erratic, agitated and unpredictable molecular motion, it is quantified through the average kinetic energy of molecules and ultimately grounded by the atomistic character of matter. Brownian motion, the shaken motion of the grains of pollen (diffusion) observed by Robert Brown in 1827, results from the bombardment of molecules experienced by the grains. The larger the average kinetic energy of such tiny colliding objects, the higher the temperature of the system. However, thermal forces are not the only way matter can be shaken, this can be also achieved by the slow release of accumulated stress in slowly relaxing systems such as structural glasses and granular media 7 , by injecting energy (for example, through external gradients) to steady-state systems 8 or through chemical reactions in self-propelled particles in active matter 9 . Can the erratic motion caused by such forces of non-thermal origin still be described in terms of the equilibriumbased concept of temperature? Spin-glass theories applied to systems exhibiting slow relaxation to equilibrium and ageing have persistently shown how the observed non-equilibrium behaviour can be described in terms of a non-equilibrium parameter, also called effective temperature. The effective temperature quantifies violations of the fluc...

For systems in an externally controllable time dependent potential, the optimal protocol minimizes the mean work spent in a finite time transition between given initial and final values of a control parameter. For an initially thermalized ensemble, we consider both Hamiltonian evolution for classical systems and Schrödinger evolution for quantum systems. In both cases, we show that for harmonic potentials, the optimal work is given by the adiabatic work even in the limit of short transition times. This result is counter-intuitive because the adiabatic work is substantially smaller than the work for an instantaneous jump. We also perform numerical calculations for the optimal protocol for Hamiltonian dynamics in an anharmonic quartic potential. For a two-level spin system, we give examples where the adiabatic work can be reached in either a finite or an arbitrarily short transition time depending on the allowed parameter space.

Controlling a time-dependent force applied to single molecules or colloidal particles is crucial for many types of experiments. Since in optical tweezers the primary controlled variable is the position of the trap, imposing a target force requires an active feedback process. We analyze this feedback process for the paradigmatic case of a nonequilibrium steady state generated by a dichotomous force protocol, first theoretically for a colloidal particle in a harmonic trap and then with both simulations and experiments for a long DNA hairpin. For the first setup, we find there is an optimal feedback gain separating monotonic from oscillatory response, whereas a too strong feedback leads to an instability. For the DNA molecule, reaching the target force requires substantial feedback gain since weak feedback cannot overcome the tendency to relax towards the equilibrium force.

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