Comparison: Is something hotter or colder; moving faster or slower? Comparison of quantity forms the most rudimentary of phenomenology, x < y 2 >1 is there a net negative or positive charge? (1.1) → ; whic π x > y 3.2 3. 2 wh ich h obje ob ject ct is fast fa ster er/h /hot otte ter/ r/co cold lder er? ? <
2 ≤ 3 splitting up into qualitative regimes ; → 2 1 y ≥ of different types of behaviour; T ≤ TC vs. T
x ≤ y x ≥
> TC
(1.2)
Fine. But what about, 0.01 >> 0.000 0.0001 1 Unsui Unsuita tabl blee for for affir affirmi ming ng stati statisti stica call hyp hypot othes heses es,, e.g e.g., ., conf confirm irming ing Higg Higgss bos boson on x >> y 0.01 ; → (1.3) Sui Su i tabl ta ble e for fo r intu in tuit itiv ive e g rasp ra sp o f p hysi hy sic c a l pict pi ctur ure e , e .g., .g ., is i t a SHO SH O w i th c orre or rect ctio ion n s? x << y 1 << 20 Orders of magnitude: what about ~ vs. <ɶ vs. >ɶ ? You’re going to use them! First consider ~,
x ~ y = y1; 1 ~ 2; 2; 10 100 ~ 30 3 00; 1 ~ 9; 9; 1 ~ 10; 10 1000 ~ 300;
(1.4)
Grey area: when two numbers are “not all that different”: 1 and 5. A factor of 5 is large, but it is not an order of magnitude. This sometimes arises in practice. 2 Example: specific heat of Fermi gas: ratio of leading term vs. next leading term is of order (k BT / E F ) .
k BT CV , F
1
3
<<
= CV , F + CV , F + ... =
EF ; n = [fermion #-density]; T
2 1 π B 2
1
k n ⋅ (kBT / EF ) 1
Example: consider another quantity, Q = Q
+O
3
= 100
K; EF / k B
00 =5
K;
3 1 2 K 2 ( kBT / EF ) → CV ,F / CV ,F ~ ( kB T / EF ) ~ ( 100 ) 500 K
2
?? 1 ) you could have Q 2 / Q1 ~ ( 100 500 ??
+ Q + ... ;
=
1 5
=
1 25
;
(1.5)
. Then, you can’t
1
use “order of magnitude reasoning”. Then, conclusions hinge upon accuracy of a measurement.
ɶ : they mean “less or of the order of”; in essence, these symbols are the respective Now let’s think about <ɶ vs. > opposites of ≫ and ≪ (note the reversed-order). Example: you could have, ɶ 2 ~ 100 , 0.1 = 10−1 ~ 10−1 <ɶ 10 = 101 100 = 1 < (1.6) Approximately equal: Finally: let’s think about the symbol ≈ (more restrictive than above), x ≈ y → x − y << z; z <ɶ x; z <ɶ y;
(1.7)
Caution: common misconception is the use of x ≈ 0 . Untrue. Suppose x = 10−5 . The problem is that you have −7 −1 8 −1234897 − −∞ ,10 910234098234 , ...,10 = 0 between x and 0. Instead, you are infinite orders of magnitude 10 ,10 ,10 supposed to write x → 0 . Example: proton vs. neutron mass: m p
= 1.672 × 10
Proportionality: this is old hat, CV , F ≈ 12 π 2 k B n ⋅ (k BT / EF )1 → CV , F
∝ T;
−27
CV ,F
kg vs. mn
∝
= 1.674 × 10
kB T / EF ; CV ,F
∝
−27
kg , so: m p
n; CV ,F
∝
− mn <<
mp .
kB 2 ← [not useful...]
(1.8)
Warning: In extracting the dependence of a physical quantity on some parameter, be careful: apparentlydifferent factors may depend on the same parameter. Example: consider a system with varying temperature, but 1
Again: conclusions are in the eye of the beholder; when things are so close, behavior changes “noticeably”, and “noticeability” is a subjective term.
fixed N (particle-number) vs. fixed µ (chemical potential). For the former case, E F while for the latter E F
= µ → CV , F ∝
n1 .
∝
n d / 2
→ CV , F ∝
n1− d /2 ,
