The metastable states of a glass are counted by adding a weak pinning field which explicitly breaks the ergodicity. Their entropy, that is the logarithm of their number, is extensive in a range of temperatures T G < T < T C only, where T G and T C correspond to the ideal calorimetric and kinetic glass transition temperatures respectively. An explicit self-consistent computation of the metastable states entropy for a non disordered model is given.

Non-deterministic polynomial time (commonly termed`NP-complete') problems are relevant to many computational tasks of practical interestÐsuch as the`travelling salesman problem'Ðbut are dif®cult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit`phase boundaries', across which dramatic changes occur in the computational dif®culty and solution characterÐthe problems become easier to solve away from the boundary. Here we report an analytic solution and experimental investigation of the phase transition in K-satis®ability, an archetypal NP-complete problem. Depending on the input parameters, the computing time may grow exponentially or polynomially with problem size; in the former case, we observe a discontinuous transition, whereas in the latter case a continuous (second-order) transition is found. The nature of these transitions may explain the differing computational costs, and suggests directions for improving the ef®ciency of search algorithms. Similar types of transition should occur in other combinatorial problems and in glassy or granular materials, thereby strengthening the link between computational models and properties of physical systems.Many computational tasks of practical interest are surprisingly dif®cult to solve even using the fastest available machines. Such problems, found for example in planning, scheduling, machine learning, hardware design, and computational biology, generally belong to the class of NP-complete problems 1±3 . NP stands for`nondeterministic polynomial time', which denotes an abstract computational model with a rather technical de®nition. Intuitively speaking, this class of computational tasks consists of problems for which a potential solution can be checked ef®ciently for correctness, yet ®nding such a solution appears to require exponential time in the worst case. A good analogy can be drawn from mathematics: proving open conjectures in mathematics is extremely dif®cult, but verifying any given proof (or solution) is generally relatively straightforward.The class of NP-complete problems lies at the foundations of the theory of computational complexity in modern computer science. Literally thousands of computational problems have been shown to be NP-complete. The completeness property of NPcomplete problems means that if an ef®cient algorithm for solving just one of these problems could be found, one would immediately have an ef®cient algorithm for all NP-complete problems. However, a fundamental conjecture of modern complexity theory is that no such ef®cient algorithm exists.Although NP-complete problems are believed to require exponential time to solve in the worst case, the typical-case behaviour is dif®cult to characterize. Yet, such typical-case properties are most relevant in practical applications. Fu and Anderson 4 ®rst conjectured a deep connection between NP-complete problems and models studied in statistical physics 5 . More recently, it was discovered (refs 6±9) that NP-compl...

In the course of evolution, proteins undergo important changes in their amino acid sequences, while their three-dimensional folded structure and their biological function remain remarkably conserved. Thanks to modern sequencing techniques, sequence data accumulate at unprecedented pace. This provides large sets of so-called homologous, i.e. evolutionarily related protein sequences, to which methods of inverse statistical physics can be applied. Using sequence data as the basis for the inference of Boltzmann distributions from samples of microscopic configurations or observables, it is possible to extract information about evolutionary constraints and thus protein function and structure. Here we give an overview over some biologically important questions, and how statistical-mechanics inspired modeling approaches can help to answer them. Finally, we discuss some open questions, which we expect to be addressed over the next years.