All of the complexity and structure we see in the world around us will ultimately be degraded and eradicated.
Consider first the fate of gravitationally bound systems, such as galaxies and galaxy clusters. Let us first recall some concepts from Newtonian gravity. Because gravitation is an attractive force, the potential energy U of a system of gravitating objects is negative; the kinetic energy K of the system, the energy of motion of the objects, is positive; and the total energy is E = U + K. A gravitating system is said to be bound if the total energy E is negative. In other words, if the magnitude of the potential energy exceeds the magnitude of the kinetic energy, the system is bound. The more negative the total energy, the more tightly bound the system is.
Each gravitating system has associated with it an escape velocity, which is the speed a constituent object must attain if the distance between it and the other objects in the system is to become unbounded. A bound system is such that the average velocity of the objects in the system is less than the escape velocity. However, after the objects in the system have interacted for a period of time, there will be a distribution of velocities, and some will exceed the escape velocity. These objects will thence depart the system, never to return. This evaporation of objects from the system will remove positive kinetic energy from the system, hence the total energy of the system will become more negative, making the system more tightly bound. Nevertheless, the system will continue to evaporate.
Contemporary spiral galaxies are bright and vibrant star cities, evolving through multiple generations of star formation, and possessing an ecological structure in which material is cycled between the population of stars, and the gas and dust of the interstellar medium. Eventually, however, star formation will cease, all the stars in a galaxy will expend their nuclear fuel, and galaxies will be populated by black holes and the dead cinders of stars. These cold, dark galaxies will then evaporate, as Iain Nicolson explains:
"Although close encounters between stars are extremely rare, given sufficient time, many encounters between dead stars will take place. In each encounter, one star will gain energy and the other will lose energy. Even without any encounters of this kind, an orbiting star will gradually lose energy by radiating gravitational waves and so, very slowly, will migrate closer to the centre of its galaxy. Close encounters will accelerate this process. Over extremely long periods of time, most dead stars will evaporate from their host galaxies and the remainder will coalesce into gigantic 'galactic' black holes at their centres. A similar process is likely to happen to clusters and superclusters of galaxies, with dead galaxies merging at their centres to form 'supergalactic' black holes, and others being ejected into intercluster space." (The End of the Universe, 1998 Yearbook of Astronomy, pp220-232).
Smaller bound systems, such as molecules and atoms will also evaporate, but the reason for this is quite subtle. As John C.Baez explains, any system in thermal equilibrium will minimise its so-called free energy, the amount of energy which is available to perform work. The free energy can be defined to be E - TS, where E is the total (internal) energy, T is the temperature of the system, and S is the entropy of the system. The restriction to internal energy here simply means that one ignores the potential energy a system might possess in an external field, and one ignores any bulk energy of motion; internal energy includes the internal potential energy of the system, and its internal kinetic energy. The entropy S of a system can be seen in this context as the amount of unusable energy in the system, per unit of system temperature; hence, multiply the entropy by the temperature, and one obtains the total amount of unusable energy in the system. Subtract the amount of unusable energy from the total energy, and one obtains the free energy.
The free energy of a system E - TS can clearly be reduced either by reducing E, or by increasing TS. As Baez points out, an ionised gas (a so-called plasma) has more energy than a gas made from atoms or molecules of the same substance. When those atoms or molecules form, electromagnetic radiation is released, decreasing the total energy of the matter in the system. However, the atoms or molecules have less entropy than the ionised system. At high temperatures, the free energy is minimised by the high entropy plasma state. However, at lower temperatures, the free energy of a matter system can be minimised by reducing the total internal energy of the system. (Note, however, that although the atomic or molecular state is a lower entropy state for the matter, the total entropy still increases because of the entropy of the electromagnetic radiation released when the atomic or molecular state is formed).
This assumes, however, that the system occupies a fixed volume. If the volume available to the constituents of the system is constantly increasing, as is the case in an expanding universe, then the maximum available entropy S of the system will be constantly increasing, and eventually, even at very low temperatures, the free energy of a gas will be minimised in the ionised, plasma state, the state which maximizes the entropy of the system.
Black holes, of course, are also capable of evaporating into radiation, but only do so if their temperature is lower than that of their surroundings. Crucially, some theorists currently argue that the presence of the dark energy, responsible for the acceleratory expansion of the universe, equips the universe with a minimum temperature. The temperature of a black hole is inversely proportional to its size, hence if sufficiently large black holes form from the merger of smaller black holes (and they would have to be as large as the currently observable universe), then such black holes would never evaporate.
Thus, (neglecting some questions over the fate of protons) the future of the universe is a future in which all galaxies, stars, planets, complex molecules and atoms eventually evaporate, and all that remains will be gravitational radiation, electromagnetic radiation, black holes, and isolated elementary particles.