"Very small" in this context is less than the lattice spacings, which for a typical crystal can be on the order of the wavelength of an X-ray (i.e., there's ~ 0.1-100 ångströms between the crystal's diffracting planes, so "distance x" must be a fraction of that length).
A typical room is effectively a <https://en.wikipedia.org/wiki/Gas_in_a_box>. If the walls of your box are good X-ray detectors then an isotropically-radiating X-ray source somewhere near the middle of the room will evidence an essentially uniform energy loss at the detectors, and and weak reflection from air molecules back towards the source. However, if you substituted the air in the room with a crystal lattice, the energy loss would be much stronger at detectors in some directions, and there would be strong reflections back towards the source along some directions. See <https://physicsopenlab.org/2018/01/18/bragg-diffraction/> for some details.
Symmetry breaking is very important to the study of condensed matter physics, including solid matter, and arguably it’s the reason your room exists at all.
https://simple.wikipedia.org/wiki/Higgs_field https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking
Uhuh, but enough to be physically detectable?
Thanks for the definition. That is close to the math idea of automophic, that is, can map onto itself. So, what physics means is that translations are essentially automorphisms. Simple now that we have a clear definition!
A spatial crystal freezes into a spatially-periodic configuration at its lowest energy: you don't need to add energy to keep a crystalline solid's microscopic components arranged in lattice-like form.
A time crystal freezes into a periodic configuration at its lowest energy: you don't need to add energy to keep a time crystal arranged in its temporally periodic arrangement. If at t_0 we have one spatial configuration, at t_1 another spatial configuration, ... at t_n-1 we have yet another spatial configuration, and at t_n we have the same spatial configuration as at t_0, and we have no net flow of energy into the spatial configuration at any t_x, we have a time crystal. The spatial configurations at any t_x need not be crystalline, they just have to differ at different points in their cycle.
An analogue clock is not a time crystal because even though the configuration of the hands at 12:00->12:01->...->11:59->12:00 is temporally periodic, you have to wind a clock (or power it in some other way) or it gets stuck at some arbitrary configuration -- it stops cycling unless "disturbed" with added energy. The clock's lowest-energy configuration has its hands always pointing to one hh:mm time, and no different time is shown over the course of a day.
A time crystal, being in its lowest-energy state, cycles through all its configurations endlessly until energy is added.
The act of "reading" the configuration of a time crystal disturbs the time crystal. So you either a set of maximally-similar time crystals that you read at various times during a day, or you need to re-freeze your single disturbed time-crystal each time you read it.
There are ordinary crystals which literally melt out of their crystalline state when handled / measured-by-bright-light. The organized pattern is broken with the additional energy. Time crystals are patterened over time, and that pattern breaks when they are handled / measured-by-bright-light.
You could think of it as having to shine a flashlight (or laser) through the time crystal to figure out which way it twists the light at a given time t_x. If you know the temporally-periodic structure, you can predict the different twisting when you turn on the light at t_x versus t_x+1 or t_x-1. But lighting up the crystal breaks the lowest-energy condition of the time crystal -- it's melted by the light it twists -- so you have to re-freeze it back into its predictable periodic structure, otherwise you might get the same twisting (or none) at t_{measured}+1, t_{measured}+2, ..., t_{measured}+n.
(It is fairly literally re-freezing: you have to do laser cooling or the like. And it takes energy to run the cooler, which removes energy from the not-lowest-energy-state broken time crystal, so thermodynamics isn't violated.)
Instead I'll direct you Sean Carroll's "Quantum Interrogation" blog posting as a starting point: <https://www.preposterousuniverse.com/blog/2006/02/27/quantum...>. ("how you can detect something without actually looking at it")
There is also the 2021 work by the Google & Stanford team (decent press release: <https://news.stanford.edu/2021/11/30/time-crystal-quantum-co...> open-access paper: <https://www.nature.com/articles/s41586-021-04257-w>, the PDF version of which (click in top right) is legible) which is at least suggestive that certain types of time crystals can be interrogated without breaking them.
Energy in.
> will keep vibrating ... lose energy slowly
Dissipative process: energy out. Eventually stop vibrating.
Time crystals cycle through their configurations until you put energy in. Otherwise, they are in lowest energy state, so no energy out. Ever.
Practically all we can do with a time crystal is to measure it with the lightest possible touch and hope it doesn't break the periodicity. (So far, afaik, the periodicity has always been broken by the measurement process's energy input).
I don't know what we could do with large numbers of time crystals, though. One can't pick up a snowflake in one's bare hands and use it like a buzz saw to cut a sheet of paper (the snowflake melts on contact), but an avalanche of snowflakes can snap trees. Maybe for time crystals that rotate light a predictable amount at a given time t_x, we could create some sort of interesting lens from a large cloud of such time crystals arranged at different distances from a bright light source -- a sort of "anti-fog".
Theory guides us, but experimentally you can for example make a whole bunch of time crystals (especially straightfoward for driven time crystals, which have a period that's an integer multiple of the driver, the driving force being laser light or microwaves) with an expected set of states it cycles through, and you can test those states once per time crystal. If you reliably get the states theory predicts, that's good evidence.
> could you outline the proof of the existence
The excellent <https://physics.aps.org/articles/v10/5> and decent <https://en.wikipedia.org/wiki/Time_crystal#Experiments> (the former is [5] in that wikipedia article) are good starting points, with ample references.
A time crystal is similar to an ordinary crystal except that instead of reducing symmetry of translations in space from a continuous symmetry to a discrete symmetry, it reduces symmetry of translations in time from a continuous symmetry to a discrete symmetry.
EDIT: It's a little ironic that if you ask most people, they would say that a crystal is more symmetric that a gas, since a gas will look completely random and asymmetric if you take a snapshot of the positions of all the atoms at a single time. But since physicists care about the probability distribution of atom positions, they say that the gas is more symmetric than the crystal.