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strange paths physics, computation, philosophy map of dark matter in the observable universe home physics computation philosophy gallery news forum about resources news: atomic orbital canon 1 a 2 xantox , 18 january 2009 in gallery other languages: français italiano in the enigmatic canon 1 a 2 from j. s. bach’s “musical offering” (1747) (also known as “crab canon” or “canon cancrizans”), the manuscript shows a single score, whose beginning joins with the end. this space is topologically equivalent to a bundle of the line segment over the circle, known as a möbius strip. the simultaneous performance of the deeply related forward and backward paths gives appearance to two voices, whose symmetry determines a reversible evolution . a musical universe is built and then is “unplayed” back into silence. 1 get the flash player to see the wordtube media player. animation created in pov-ray by jos leys . music performed by xantox with post flemish harpsichord, upper manual. [ ↩ ] 7 comments » share this atomic orbital xantox , 20 april 2008 in gallery other languages: français italiano time evolution of an hydrogenic (single-electron) atomic orbital with quantum numbers | 3, 2, 1 > according to the schrödinger equation (colors represent phase). in atomic matter, electrons orbiting the nucleus do not follow any determined classical path, but exist for each quantum state within an orbital, which can be visualized as a cloud of the probabilities of observing the electron at any given location and time. get the flash player to see the wordtube media player. click image to zoom 1 © dean e. dauger [ ↩ ] comments » share this reversible computation xantox , 20 january 2008 in computation other languages: français italiano a computation (from latin computare, “to count”, “to cut”), is the abstract representation of a physical process in terms of states , and transitions between states or events . the definition of possible states and events is formulated in a computation model , such as the turing machine or the finite automaton. for example, a turing machine state is the complete sequence of symbols on its tape plus the head’s position and internal symbol, and an event is the motion between two successive states, defined deterministically as a combination of read, write, move left and move right elementary motions. in order to perform a computation, a robust mapping is first established between a computation model and a physical system, meaning that states and events in the model are used to label states and events observed in the system, and that the choosen correspondence is sufficiently stable in respect to various kinds of perturbations. the system is then prepared in an initial state and is allowed to evolve through a path of events within the space of states, until it eventually reaches a state labeled as final. the discretized dynamics of the computational space may be represented with a directed graph, where nodes are possible states of the system and edges are events transforming a state into another. irreversible computational dynamics click image to zoom 1 logical reversibility a function is said reversible (from latin revertere, ‘to turn back’) if, given its output, it is always possible to determine back its input, which is the case when there is a one-to-one relationship between input and output states. if the space of states is finite, such a function is a permutation. logical reversibility implies conservation of information. when several input states are mapped onto the same output state, then the function is irreversible, since it is impossible by only knowing the final state to find back the initial state. in boolean algebra, not is reversible, while set to one is irreversible. two-argument boolean functions like and, or, xor are also irreversible, since they map 2 2 input states into 2 1 output states so that information is lost in the merging of paths, like shown in the following graph of a nand computation, whose reverse evolution is no longer deterministic. the right side tries to depict the inverse mapping to the left side physical reversibility known laws of physics are reversible. this is the case both of classical mechanics, based on lagrangian/hamiltonian dynamics, and of standard quantum mechanics, where closed systems evolve by unitary transformations, which are bijective and invertible. as a consequence, when a physical system performs an irreversible computation, the computation model’s mapping indicates that the computing system cannot stay closed. more precisely, since an irreversible computation reduces the space of physical information-bearing states, then their entropy must decrease by increasing the entropy of the non-information bearing states, representing the thermal part of the system. in 1961 landauer studied this thermodynamical argument, and proposed the following principle: if a physical system performs a logically irreversible classical computation, then it must increase the entropy of the environment with an absolute minimum of heat release of kt x ln(2) per lost bit (where k is boltzmann’s constant and t the temperature, ie. about 3 x 10 -21 joules at room temperature), 2 which emphasizes two facts: the logical irreversibility of a computation implies the physical irreversibility of the system performing it (”information is physical”); logically reversible computations may be at least in principle intrinsically nondissipative (which bears a relationship with carnot’s heat engine theorem, showing that the most efficient engines are reversible ones, and clausius theorem, attributing zero entropy change to reversible processes). reversible embedding of irreversible computations landauer further noticed that any irreversible computation may be transformed into a reversible one by embedding it into a larger computation where no information is lost, eg. by replicating every output in the input (’sources’) and every input in the output (’sinks’). for example, the nand irreversible function seen above may be embedded in the following bijection, also known as toffoli gate 3 (the original function is indicated in red): the additional bits of information, like ariadne’s threads, ensure that any computational path may be reversed: they are the garbage of the forward path and the program of the backwards path. instead of losing them in the environment, they are kept in the controlled computational space. toffoli gates are universal reversible logic primitives, meaning that any reversible function may be constructed in terms of toffoli gates. the fredkin gate is another example of universal reversible logic primitive. it exchanges its two inputs depending on the state of a third control input, thus allowing to embed any computation into a conditional routing of paths carrying conserved signals. some railroad switches are reversible reversible computation models the billiard-ball model, invented by fredkin and toffoli, 4 was one of the first computation models focusing on implementation with reversible physical components. based on the laws of classical mechanics, it is equivalent to the formalism of kinetic theory of perfect gases. the presence of moving rigid spheres at specified points are defined as 1’s, their absence as 0’s. interactions by means of right-angle collisions allow to construct various logic primitives, like for example the following 2-input, 3-output universal gate due to feynman, 5 who also proposed with ressler a billiard-ball version of the fredkin gate. feynman switch gate b detects a without affecting its path in practice, these computing spheres would be very unreliable, as instability arising from arbitrarily small perturbations would quickly generate chaotic deviations, producing an output saturated with errors. the errors may be corrected (for example, by adding potentials to stabilize the paths), however the error correction process is itself irreversible and dissipative - since it has to erase the

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