1. INTRODUCTION

Cosmological simulations of the large-scale structure (LSS) predict that about 40-50% of baryons at epoch z < 2 could reside in the Warm-Hot Intergalactic Medium (WHIM) with temperatures 10⁵ - 10⁷ K at moderate overdensities < 100 (Cen & Ostriker 1999, Davé et al. 2001, Fang et al. 2002). The WHIM heating is due to shocks driven by gravitationally accelerated flows in the LSS structure formation scenario (e.g. Kang et al. 2007). Numerical simulations predict the observational signatures of the web gas as a function of redshift. The simulations account for feedback interactions between galaxies and the intergalactic medium, and demonstrate that the X-ray and ultraviolet O VI, O VII and O VIII lines and the H I Lyman alpha line are good tracers of low-density cosmic web filamentary structures (e.g. Hellsten et al. 1998, Tripp et al. 2000, Cen et al. 2001, Furlanetto et al. 2004). Intervening metal absorption systems of highly ionised C, N, O, Ne in the soft X-ray spectra of bright Active Galactic Nuclei (AGN) were suggested to be tracer of the WHIM. The predicted distribution of ion column densities in the WHIM absorbers is steep enough to provide only a few systems with N_{O VII} > 10¹⁵ cm^-2 along an arbitrary chosen line of sight (e.g. Fang et al. 2002). Therefore, the detection of the WHIM is particularly difficult and requires very sensitive UV and X-ray detectors, both for absorption and for emission processes (e.g. Lehner et al. 2007, Nicastro et al. 2005, Kaastra et al. 2006, Takei et al. 2007 and Richter et al. 2008, Durret et al. 2008 - Chapters 3 and 4, this volume). Future projects and namely Cosmic Origin Spectrograph (COS), the X-Ray Evolving Universe Spectrometer (XEUS), Constellation-X and the Diffuse Intergalactic Oxygen Surveyor (DIOS) will increase the signal-to-noise ratio in the spectra allowing to study weak systems with N_{H I} < 10^12.5 cm^-2 and N_{O VII} < 10¹⁵ cm^-2. Simulations of spectra of the broad Ly absorption lines and the highly ionised oxygen lines in the weak systems require thorough modelling of physical condition in the plasma (see e.g. Mewe 1990, Paerels & Kahn 2003, Kawahara et al. 2006). We discuss in this paper the heating and equilibration processes in the shocked WHIM plasma affecting the spectral simulations. A discussion of collisionless shock physics relevant to cosmological shocks in the WHIM can be found in Bykov et al. (2008) - Chapter 7, this volume. To this end, in this paper we discuss first some specific features of collisionless shock heating of ions of different charge states providing highly non-equilibrium initial states just behind the magnetic ramp region that relaxes to an equilibrium state through Coulomb collisions, and the relation of the processes to simulations of emission/absorption spectra of the WHIM and observational data analysis.