What's the difference between "дети бегают по магазину" and "дети бегают в магазине"?

Question

What's the difference between "дети бегают по магазину" and "дети бегают в магазине"?

EDIT: If you are about to provide some examples in your answers, please, observe the following requests:

1. consider only those examples that contain a motion verb (like прыгают, танцуют, летают, etc. - not the verbs like думают, мечтают, находятся, etc.);

2. avoid using unidirectional verbs (like бегут, плывут, летят, едут, несутся, etc.), use their multidirectional counterparts instead (бегают, плавают, летают, ездят, носятся, etc.)

3. the noun following the proposition "в" should be in the propositional case ("ходим в магазине"), not in the accusative case ("сходим в магазин")

Answer 1

The two meanings are very close, with the "по" hinting at the randomness of the running, while the "в" stressing the location.

My translations of the two phrases to English would be as follows:

Kids are running inside the store. ("в магазине")

Kids are running around the store. ("по магазину")

Answer 2

Rosenthal et al., СППЛР, 199.3

Смысловые различия находим внутри каждой из приводимых ниже пар:

• гулять в лесу – гулять по лесу: первое сочетание обозначает действие ограниченное (гулять можно на небольшом участке леса), а второе – действие разбросанное (в пределах названного пространства);

which means:

Each of the following pairs exposes semantic differences:

• гулять в лесу – гулять по лесу: the first phrase defines action confined in space (one can walk in a smaller part of the forest), the second one defining a scattered action (within the limits of the named enclosure)

This difference is more clear if we consider a spatially anisotropic object:

• идти на улице / идти по улице
• плыть на реке / плыть по реке

The first sentence may (or may not) suggest that you cross the street or the river, the second sentence strictly assumes that you move along the street or the river.

However, which is more important, is that по unequivocally makes the noun the direct object of the verb, while в / на may imply an adverbial clause будучи:

Compare:

• Спортсмены бегают в зале // The sportsmen are running in the gym
• Спортсмены бегают по залу // The sportsmen are running over the gym.

The first sentence may imply that they are, say, running on the treadmills, while being in the gym.

The second sentence cannot have this meaning.

Answer 3

Use of по assumes the importance, in the context, of the spatial extent of the place. Быть в магазине simply means that a person is within a shop and nowhere else. The shop is considered a point in space. Ходить по магазину assumes that the shop has some extent and one needs some time to traverse it. Compare also:

Пожалуйста, сходи в магазин.

and

Давай пройдёмся по магазинам.

The first sentence emphasizes the way to the shop while the second one the process of shopping, being in the shops or going from one to another.

Answer 4

I'm not sure but I think there is no much difference. Although probably "дети бегают в магазине" is to say where the children are (in the shop), while "дети бегают по магазину" is more like to say what the children are doing (running) and, especially, doing something wrong or unpleasant. Like "Я бегаю по магазину в поисках чего-либо" means that I going round between shop desks to find something. Also "не бегай по коридору" would be used to stop a child from running at all, while "не бегай в коридоре" is better to ask a child to find another place than corridor for running in (like street or so on).