The Hamiltonian is a concept in (unsurprisingly) in Hamiltonian mechanics which represents an operator in a system. In many cases it corresponds to the total energy of a system and so I’ll continue to use it as analogous for the total energy although this not always true. Just picture it as an equation with each term representing one of the kinds of energy a system contains. Now often you want to find the ground state, the lowest possible energy, for the system as this will provide useful information to solve it. But what if this Hamiltonian’s ground state is very complicated, if not impossible, to find algebraically? Well, this is the basis of an adapted form of quantum computing called adiabatic quantum computing.
There are many occasions in engineering where it is desired for a material to absorb light. Thermal photovoltaic cells, for instance, want to try to absorb as much energy as they can. Any reflection that occurs off a surface is a loss of energy they could have been using and so represents and inherent limit on the efficiency of these cells. In a way this is also true for devices that produce heat. Any of the produced heat from an electric circuit that can be recaptured and reused will overall drastically increase the efficiency of circuits as thermal loss is the man source of inefficiency.
When you have a bunch of particles in a system moving and interacting with each other, what would happen if you began to change the density or temperature of the particles. It’s quite intuitive to see as the particle density increases that the whole system approaches that of an amorphous solid as the particles begin to be crammed together. Depending on the nature of the interacting forces between the particles, the same thing can happen when the temperature is dropped. Even systems made up of hard spheres have been seen to undergo this glass transition, in fact I have even written about it before in this post.
I think its come the point where the qualities of graphene do not even need to be stated. The very fact that there almost one paper published featuring it everyday in Nature goes to show that its discovery truly was worthy of a Nobel prize. Over the years various improvements in manufacturing techniques means that we don’t need to use Sellotape and pencil graphite to produce graphene any more. Vapour deposition and ultrasonification allow for relatively cheap production of 2D graphene for a variety of applications.
The simple fact is that sometimes there is no solution to a problem. This is not a response that many people would like to hear but it is an unfortunate truth. A particularly famous example would be the three body problem, where the equations of motion for three masses with arbitrary starting positions and velocities are simply unsolvable. In many ways it is the balance that scientific models strive for to both be accurate to physical reality but also solvable. If it impossible to do both then making a model accurate but solvable by numerical methods is acceptable. The worst situation is that to get an accurate model will require creating the model so that even a computer will struggle with providing solutions as complexity grows.
Thermodynamics is quite an incredible subject. Originally designed in order to understand the maximum amount of mechanical work that could be gained from a steam engine it now has branches spreading all over the study of physics. Despite being derived from mechanics it managed to remain unchanged even with the quantum and relativistic revolutions. The principles of thermodynamics were originally stated in three laws:
- Total energy is a constant in a closed system although energy can be converted by work into heat.
- Heat will always have a net flow from a hotter to a cooler place, resulting in thermal equilibrium for a closed system. This is the equivalent of saying that net entropy always increases.
- The entropy of an object approaches a constant value (often very close to zero) as the temperature falls to absolute zero.
After all these were made the zeroth law was added which states that if object A and object B are at the same temperature; and object B and object C are at the same temperature than object A and object C must be as the same temperature. Although this seems obvious it is a necessary definition of what temperature and thermal equilibrium are.
Charles Darwin is of course famous for his suggestion that it is competition and natural selection that result in the evolution of species. The observation bias made manifest, only those things which survive get seen by us and everything that failed, died. Of course back in the nineteenth century it was not known how these changes arose, it was only with the advent of genetics that the idea of random mutations driving the changes became the standard. The other main idea now accepted is that of genetic drift. It isn’t necessary for animals to be constantly under threat for evolution to occur, over time, chance will simply make some genetic traits develop and remain while others will be lost whether they be positive, negative or neutral. Nowadays we can study evolution experimentally where we can view a chosen species over many generations and by controlling the laboratory conditions can observe their evolution. This would take considerable time if the species chosen was an animal or plant and so instead it is normally rapidly reproducing bacteria that are the focus.